💡Okay, so I have this problem with a circle and a line, and I need to figure out which of the statements A, B, C, and D are correct. Let me start by understanding what each statement is saying.First, the circle is given by the equation ( x^2 + y^2 = 1 ). That's a standard equation for a circle centered at the origin (0,0) with a radius of 1. Got that. The line is given by ( y = x + 1 ). That's a straight line with a slope of 1 and a y-intercept at (0,1). Let's go through each option one by one.**Option A: The intercept of line ( l ) on the y-axis is 1.**Hmm, the y-intercept is the point where the line crosses the y-axis. For the line ( y = x + 1 ), when ( x = 0 ), ( y = 0 + 1 = 1 ). So, the y-intercept is indeed at (0,1). That means the intercept is 1. So, A is correct.**Option B: The slope of line ( l ) is ( frac{pi}{4} ).**Wait, the slope of a line is usually a number, not an angle. The slope here is 1 because the equation is ( y = x + 1 ). But ( frac{pi}{4} ) is an angle, specifically 45 degrees. Is there a connection here? Oh, right! The slope is equal to the tangent of the angle that the line makes with the positive x-axis. So, if the slope is 1, then ( tan(theta) = 1 ), which means ( theta = frac{pi}{4} ). So, the angle is ( frac{pi}{4} ), but the slope itself is 1, not ( frac{pi}{4} ). So, is B correct? It says the slope is ( frac{pi}{4} ), which is not accurate because the slope is 1. Maybe it's trying to say the angle is ( frac{pi}{4} ), but the wording is about the slope. So, I think B is incorrect.**Option C: Line ( l ) intersects circle ( C ) at 2 points.**To check this, I need to see if the line intersects the circle. One way to do this is to solve the system of equations:1. ( x^2 + y^2 = 1 )2. ( y = x + 1 )Substitute equation 2 into equation 1:( x^2 + (x + 1)^2 = 1 )Expanding that:( x^2 + x^2 + 2x + 1 = 1 )Combine like terms:( 2x^2 + 2x + 1 = 1 )Subtract 1 from both sides:( 2x^2 + 2x = 0 )Factor out 2x:( 2x(x + 1) = 0 )So, ( x = 0 ) or ( x = -1 )Now, find the corresponding y-values:If ( x = 0 ), then ( y = 0 + 1 = 1 ). So, one point is (0,1).If ( x = -1 ), then ( y = -1 + 1 = 0 ). So, the other point is (-1,0).Therefore, the line intersects the circle at two points: (0,1) and (-1,0). So, C is correct.**Option D: The maximum distance from a point on circle ( C ) to line ( l ) is ( sqrt{2} ).**Hmm, this one is a bit trickier. I need to find the maximum distance from any point on the circle to the line ( y = x + 1 ). First, I recall that the distance from a point ( (x_0, y_0) ) to the line ( ax + by + c = 0 ) is given by:( d = frac{|ax_0 + by_0 + c|}{sqrt{a^2 + b^2}} )Let me rewrite the line equation in standard form:( y = x + 1 ) can be rewritten as ( x - y + 1 = 0 ). So, ( a = 1 ), ( b = -1 ), and ( c = 1 ).Now, the distance from a point ( (x, y) ) on the circle to the line is:( d = frac{|x - y + 1|}{sqrt{1^2 + (-1)^2}} = frac{|x - y + 1|}{sqrt{2}} )Since the circle is ( x^2 + y^2 = 1 ), any point on the circle can be represented as ( ( cos theta, sin theta ) ) where ( theta ) is the angle parameter.Substituting ( x = cos theta ) and ( y = sin theta ) into the distance formula:( d = frac{|cos theta - sin theta + 1|}{sqrt{2}} )Now, to find the maximum distance, I need to maximize the numerator ( |cos theta - sin theta + 1| ).Let me consider the expression ( cos theta - sin theta ). I know that ( cos theta - sin theta ) can be rewritten using the identity:( cos theta - sin theta = sqrt{2} cos left( theta + frac{pi}{4} right) )So, substituting back:( |sqrt{2} cos left( theta + frac{pi}{4} right) + 1| )Therefore, the distance becomes:( d = frac{|sqrt{2} cos left( theta + frac{pi}{4} right) + 1|}{sqrt{2}} )To find the maximum value of ( d ), I need to find the maximum of ( |sqrt{2} cos left( theta + frac{pi}{4} right) + 1| ).The maximum value of ( cos ) function is 1, so:Maximum ( |sqrt{2} cdot 1 + 1| = |sqrt{2} + 1| = sqrt{2} + 1 )Therefore, the maximum distance is:( d_{text{max}} = frac{sqrt{2} + 1}{sqrt{2}} = 1 + frac{1}{sqrt{2}} approx 1 + 0.707 = 1.707 )But the statement says the maximum distance is ( sqrt{2} approx 1.414 ), which is less than 1.707. So, the maximum distance is actually greater than ( sqrt{2} ). Therefore, D is incorrect.Wait, but maybe I made a mistake somewhere. Let me double-check.Alternatively, another approach is to consider the distance from the center of the circle to the line and then add the radius.The distance from the center (0,0) to the line ( y = x + 1 ) is:( d = frac{|0 - 0 + 1|}{sqrt{1^2 + (-1)^2}} = frac{1}{sqrt{2}} approx 0.707 )Since the radius of the circle is 1, the maximum distance from any point on the circle to the line would be ( d + r = frac{1}{sqrt{2}} + 1 approx 1.707 ), which is the same as before.So, the maximum distance is indeed ( 1 + frac{1}{sqrt{2}} ), which is approximately 1.707, not ( sqrt{2} approx 1.414 ). Therefore, D is incorrect.Wait, but maybe I'm misunderstanding the question. It says "the maximum distance from a point on circle C to line l is ( sqrt{2} )". Maybe it's referring to the maximum distance as the distance from the farthest point on the circle to the line, which could be the distance from the center plus the radius in the direction away from the line.But in this case, the distance from the center to the line is ( frac{1}{sqrt{2}} ), and the radius is 1. So, the maximum distance would be ( frac{1}{sqrt{2}} + 1 ), which is about 1.707, not ( sqrt{2} ).Alternatively, maybe it's considering the diameter? The diameter of the circle is 2, but that's not directly related to the distance to the line.Wait, another way to think about it is to find the point on the circle that is farthest from the line. The distance from the center to the line is ( frac{1}{sqrt{2}} ), and the maximum distance would be this distance plus the radius, but in the direction away from the line. So, it's ( frac{1}{sqrt{2}} + 1 ), which is approximately 1.707.But ( sqrt{2} ) is approximately 1.414, which is less than 1.707. So, the maximum distance is actually greater than ( sqrt{2} ). Therefore, D is incorrect.Wait, but maybe I'm overcomplicating it. Let me try another approach. The maximum distance from a point on the circle to the line is the distance from the center to the line plus the radius times the maximum value of the directional derivative or something like that.Alternatively, maybe I can parametrize the point on the circle and find the maximum distance.Let me consider the point ( (x, y) ) on the circle ( x^2 + y^2 = 1 ). The distance to the line ( y = x + 1 ) is ( frac{|x - y + 1|}{sqrt{2}} ).To maximize this, I can set up the function ( f(x, y) = frac{|x - y + 1|}{sqrt{2}} ) subject to the constraint ( x^2 + y^2 = 1 ).Using Lagrange multipliers, I can set up the system:( nabla f = lambda nabla g )Where ( g(x, y) = x^2 + y^2 - 1 = 0 ).But since we have an absolute value, it's a bit more complicated. Maybe I can consider the square of the distance to avoid the absolute value.Let me define ( f(x, y) = (x - y + 1)^2 ). Then, to maximize ( f(x, y) ) subject to ( x^2 + y^2 = 1 ).Using Lagrange multipliers:( nabla f = lambda nabla g )Compute gradients:( nabla f = [2(x - y + 1), -2(x - y + 1)] )( nabla g = [2x, 2y] )So, setting up the equations:1. ( 2(x - y + 1) = lambda 2x )2. ( -2(x - y + 1) = lambda 2y )3. ( x^2 + y^2 = 1 )From equation 1:( x - y + 1 = lambda x )From equation 2:( -(x - y + 1) = lambda y )Let me call equation 1 as:( x - y + 1 = lambda x ) → ( x(1 - lambda) - y + 1 = 0 )And equation 2 as:( -(x - y + 1) = lambda y ) → ( -x + y - 1 = lambda y ) → ( -x + y(1 - lambda) - 1 = 0 )Now, I have two equations:1. ( x(1 - lambda) - y + 1 = 0 )2. ( -x + y(1 - lambda) - 1 = 0 )Let me write them as:1. ( (1 - lambda)x - y = -1 )2. ( -x + (1 - lambda)y = 1 )This is a system of linear equations in x and y. Let me write it in matrix form:[begin{bmatrix}1 - lambda & -1 -1 & 1 - lambdaend{bmatrix}begin{bmatrix}x yend{bmatrix}=begin{bmatrix}-1 1end{bmatrix}]To solve this system, I can compute the determinant of the coefficient matrix:( D = (1 - lambda)^2 - (-1)(-1) = (1 - lambda)^2 - 1 )For non-trivial solutions, D ≠ 0, but since we are looking for specific solutions, let's proceed.Let me solve for x and y.From equation 1:( (1 - lambda)x - y = -1 ) → ( y = (1 - lambda)x + 1 )Substitute y into equation 2:( -x + (1 - lambda)y = 1 )Substitute y:( -x + (1 - lambda)[(1 - lambda)x + 1] = 1 )Expand:( -x + (1 - lambda)^2 x + (1 - lambda) = 1 )Combine like terms:( [ -1 + (1 - lambda)^2 ]x + (1 - lambda) = 1 )Let me compute ( (1 - lambda)^2 - 1 ):( (1 - 2lambda + lambda^2) - 1 = -2lambda + lambda^2 )So, the equation becomes:( (-2lambda + lambda^2)x + (1 - lambda) = 1 )Now, let's solve for x:( (lambda^2 - 2lambda)x = 1 - (1 - lambda) = lambda )So,( x = frac{lambda}{lambda^2 - 2lambda} = frac{lambda}{lambda(lambda - 2)} = frac{1}{lambda - 2} )Now, substitute x back into y = (1 - λ)x + 1:( y = (1 - lambda)left( frac{1}{lambda - 2} right) + 1 = frac{1 - lambda}{lambda - 2} + 1 )Simplify:( y = frac{1 - lambda + lambda - 2}{lambda - 2} = frac{-1}{lambda - 2} )So, we have:( x = frac{1}{lambda - 2} )( y = frac{-1}{lambda - 2} )Now, recall that ( x^2 + y^2 = 1 ):( left( frac{1}{lambda - 2} right)^2 + left( frac{-1}{lambda - 2} right)^2 = 1 )Simplify:( frac{1}{(lambda - 2)^2} + frac{1}{(lambda - 2)^2} = 1 )Combine:( frac{2}{(lambda - 2)^2} = 1 )Multiply both sides by ( (lambda - 2)^2 ):( 2 = (lambda - 2)^2 )Take square roots:( lambda - 2 = pm sqrt{2} )So,( lambda = 2 pm sqrt{2} )Now, let's find x and y for each λ.**Case 1: ( lambda = 2 + sqrt{2} )**( x = frac{1}{(2 + sqrt{2}) - 2} = frac{1}{sqrt{2}} )( y = frac{-1}{sqrt{2}} )So, the point is ( left( frac{1}{sqrt{2}}, -frac{1}{sqrt{2}} right) )**Case 2: ( lambda = 2 - sqrt{2} )**( x = frac{1}{(2 - sqrt{2}) - 2} = frac{1}{- sqrt{2}} = -frac{1}{sqrt{2}} )( y = frac{-1}{- sqrt{2}} = frac{1}{sqrt{2}} )So, the point is ( left( -frac{1}{sqrt{2}}, frac{1}{sqrt{2}} right) )Now, let's compute the distance from these points to the line ( y = x + 1 ).First, for ( left( frac{1}{sqrt{2}}, -frac{1}{sqrt{2}} right) ):( d = frac{|frac{1}{sqrt{2}} - (-frac{1}{sqrt{2}}) + 1|}{sqrt{2}} = frac{|frac{2}{sqrt{2}} + 1|}{sqrt{2}} = frac{|sqrt{2} + 1|}{sqrt{2}} = frac{sqrt{2} + 1}{sqrt{2}} = 1 + frac{1}{sqrt{2}} approx 1.707 )For ( left( -frac{1}{sqrt{2}}, frac{1}{sqrt{2}} right) ):( d = frac{|-frac{1}{sqrt{2}} - frac{1}{sqrt{2}} + 1|}{sqrt{2}} = frac{|-frac{2}{sqrt{2}} + 1|}{sqrt{2}} = frac{|- sqrt{2} + 1|}{sqrt{2}} )Since ( sqrt{2} approx 1.414 ), ( -sqrt{2} + 1 approx -0.414 ), so the absolute value is ( sqrt{2} - 1 ).Thus,( d = frac{sqrt{2} - 1}{sqrt{2}} = 1 - frac{1}{sqrt{2}} approx 0.293 )So, the maximum distance is ( 1 + frac{1}{sqrt{2}} approx 1.707 ), which is greater than ( sqrt{2} approx 1.414 ). Therefore, D is incorrect.Wait, but maybe I made a mistake in interpreting the maximum distance. Let me think again.Alternatively, the maximum distance could be the distance from the center plus the radius in the direction perpendicular to the line. The distance from the center to the line is ( frac{1}{sqrt{2}} ), and the radius is 1. So, the maximum distance would be ( frac{1}{sqrt{2}} + 1 approx 1.707 ), which is indeed greater than ( sqrt{2} ).Therefore, the maximum distance is not ( sqrt{2} ), so D is incorrect.Wait, but maybe the question is referring to the maximum distance as the distance from the center plus the radius in the direction of the line, but that doesn't make sense because the line is not a vector.Alternatively, maybe the maximum distance is the distance from the center to the line plus the radius times the maximum value of the directional derivative, but I think that's more complicated and not necessary here.Given all this, I think D is incorrect because the maximum distance is actually ( 1 + frac{1}{sqrt{2}} ), which is approximately 1.707, not ( sqrt{2} ).So, summarizing:- A: Correct- B: Incorrect (slope is 1, not ( frac{pi}{4} ))- C: Correct- D: IncorrectTherefore, the correct options are A and C.Wait, but earlier I thought B was incorrect because the slope is 1, not ( frac{pi}{4} ). But maybe the question is referring to the angle of inclination, which is ( frac{pi}{4} ). So, if B says the slope is ( frac{pi}{4} ), that's incorrect because the slope is 1, but the angle is ( frac{pi}{4} ). So, B is incorrect.So, final answer: A and C are correct.

💡Okay, so I have this problem here: A rectangle with integer length and width has a perimeter of 200 units. I need to find the number of square units in the least possible area that is divisible by 10. Hmm, let me try to figure this out step by step.First, I remember that the perimeter of a rectangle is calculated by the formula P = 2*(length + width). In this case, the perimeter is 200 units. So, if I let the length be 'x' and the width be 'y', then the equation would be:2*(x + y) = 200If I simplify that, I divide both sides by 2:x + y = 100So, that means the sum of the length and the width is 100. Since both x and y are integers, I can express y in terms of x:y = 100 - xOkay, so now I can express the area of the rectangle, which is A = x * y, in terms of x:A = x*(100 - x) = 100x - x^2So, the area is a quadratic function of x, and it's a downward-opening parabola because the coefficient of x^2 is negative. That means the maximum area occurs at the vertex of the parabola, but I'm interested in the minimum area that's divisible by 10.Wait, the problem says "the least possible area that is divisible by 10." So, I need to find the smallest area that is a multiple of 10. Since the area is given by A = 100x - x^2, I need to find the smallest value of A that is divisible by 10.But how do I approach this? Maybe I should consider the possible integer values of x and y that satisfy x + y = 100 and then calculate the area for each pair to find the smallest one divisible by 10.Since x and y are positive integers, x can range from 1 to 99. But checking all 99 possibilities seems tedious. Maybe there's a smarter way.I know that for a number to be divisible by 10, it must end with a 0. So, the area A must end with a 0. That means A mod 10 = 0.Given A = 100x - x^2, let's see what this expression mod 10 looks like:A mod 10 = (100x - x^2) mod 10But 100x mod 10 is 0 because 100 is a multiple of 10. So, A mod 10 = (-x^2) mod 10Which simplifies to:A mod 10 = (10 - (x^2 mod 10)) mod 10Wait, that might not be the easiest way to think about it. Let me consider x^2 mod 10.The possible values of x^2 mod 10 are limited because the last digit of x determines the last digit of x^2. For example:0^2 = 0 mod 101^2 = 1 mod 102^2 = 4 mod 103^2 = 9 mod 104^2 = 16 mod 10 = 65^2 = 25 mod 10 = 56^2 = 36 mod 10 = 67^2 = 49 mod 10 = 98^2 = 64 mod 10 = 49^2 = 81 mod 10 = 1So, the possible values of x^2 mod 10 are 0,1,4,5,6,9.Therefore, A mod 10 = (-x^2) mod 10 = (10 - (x^2 mod 10)) mod 10So, if x^2 mod 10 is 0, then A mod 10 is 0If x^2 mod 10 is 1, then A mod 10 is 9If x^2 mod 10 is 4, then A mod 10 is 6If x^2 mod 10 is 5, then A mod 10 is 5If x^2 mod 10 is 6, then A mod 10 is 4If x^2 mod 10 is 9, then A mod 10 is 1So, A mod 10 is 0 only when x^2 mod 10 is 0, which happens when x mod 10 is 0. So, x must be a multiple of 10.Therefore, x must be 10, 20, 30, ..., 90.So, the possible x values are 10,20,...,90.Now, I need to find the corresponding areas for these x values and find the smallest one.Let me calculate A for x=10:A = 100*10 - 10^2 = 1000 - 100 = 900For x=20:A = 100*20 - 20^2 = 2000 - 400 = 1600x=30:A=100*30 - 30^2=3000-900=2100x=40:A=4000-1600=2400x=50:A=5000-2500=2500x=60:A=6000-3600=2400x=70:A=7000-4900=2100x=80:A=8000-6400=1600x=90:A=9000-8100=900So, the areas for x=10,20,...,90 are 900,1600,2100,2400,2500,2400,2100,1600,900.Looking at these, the smallest area is 900, which occurs at x=10 and x=90.But wait, is 900 the smallest possible area divisible by 10? Let me check if there are any smaller areas that are divisible by 10 but not necessarily at x being a multiple of 10.Earlier, I concluded that x must be a multiple of 10 for A to be divisible by 10. But let me verify that.Suppose x is not a multiple of 10, but A is still divisible by 10. Is that possible?From earlier, A mod 10 = (10 - (x^2 mod 10)) mod 10For A mod 10 to be 0, (10 - (x^2 mod 10)) mod 10 must be 0, which implies x^2 mod 10 must be 0.Therefore, x must be a multiple of 10. So, my initial conclusion was correct.Therefore, the smallest area divisible by 10 is 900.Wait, but let me think again. Maybe I can find a smaller area that is divisible by 10 without x being a multiple of 10.Suppose x=5, then y=95.A=5*95=475, which is not divisible by 10.x=15, y=85A=15*85=1275, which is divisible by 5 but not by 10.x=25, y=75A=25*75=1875, which is divisible by 5 but not by 10.x=35, y=65A=35*65=2275, same as above.x=45, y=55A=45*55=2475x=55, y=45Same as above.x=65, y=35Same as above.x=75, y=25Same as above.x=85, y=15Same as above.x=95, y=5Same as above.So, none of these areas are divisible by 10. Therefore, my initial conclusion holds: the smallest area divisible by 10 is 900.But wait, let me check x=10 and x=90. For x=10, y=90, area=900. For x=90, y=10, same area.Is there a smaller x that is a multiple of 10? The smallest x can be is 10, since x=0 would make the rectangle degenerate.Therefore, 900 is indeed the smallest area divisible by 10.Wait, but let me think about the area function again. The area is A = 100x - x^2. This is a quadratic function that opens downward, with its maximum at x=50, where A=2500.As x moves away from 50 towards 0 or 100, the area decreases. So, the minimal areas occur at the extremes, x=1 and x=99, but those areas are 99 and 99, which are not divisible by 10.But the next possible minimal areas that are divisible by 10 would be at x=10 and x=90, giving A=900.Therefore, the least possible area divisible by 10 is 900 square units.

💡First, I'll list out all the given information:1. Neither Anne nor Dan has as much money as Bo.2. Both Anne and El have more money than Coe.3. Dan has more money than Coe but less than Anne.Next, I'll analyze each piece of information to establish relationships between the individuals:- From the first point, Bo has more money than both Anne and Dan.- From the second point, Coe has less money than both Anne and El.- From the third point, Dan has more money than Coe but less than Anne.Now, I'll deduce who has the least amount of money:- Bo cannot have the least because he has more money than Anne and Dan.- Anne and El cannot have the least because they both have more money than Coe.- Dan cannot have the least because he has more money than Coe.The only remaining person is Coe, who has less money than Anne, El, and Dan. Therefore, Coe has the least amount of money.

💡First, I need to determine the values of (cos alpha) and (sin alpha) based on the coordinates of point (Pleft(frac{3}{5}, -frac{4}{5}right)).The x-coordinate of point (P) is (frac{3}{5}), which corresponds to (cos alpha). Therefore:[cos alpha = frac{3}{5}]The y-coordinate of point (P) is (-frac{4}{5}), which corresponds to (sin alpha). Therefore:[sin alpha = -frac{4}{5}]Next, I need to calculate (cos alpha - sin alpha):[cos alpha - sin alpha = frac{3}{5} - left(-frac{4}{5}right) = frac{3}{5} + frac{4}{5} = frac{7}{5}]Thus, the value of (cos alpha - sin alpha) is (frac{7}{5}).

💡Okay, so I have this matrix:[begin{pmatrix} a & frac{15}{34} c & frac{25}{34} end{pmatrix}]And it's supposed to be a projection matrix. Hmm, I remember that projection matrices have a special property where if you multiply the matrix by itself, you get the same matrix back. So, mathematically, that means:[mathbf{P}^2 = mathbf{P}]Where (mathbf{P}) is our given matrix. So, I need to compute (mathbf{P}^2) and set it equal to (mathbf{P}), then solve for (a) and (c).Alright, let's compute (mathbf{P}^2). To do that, I'll multiply the matrix by itself:[mathbf{P}^2 = begin{pmatrix} a & frac{15}{34} c & frac{25}{34} end{pmatrix} times begin{pmatrix} a & frac{15}{34} c & frac{25}{34} end{pmatrix}]Let me recall how matrix multiplication works. Each element in the resulting matrix is the dot product of the corresponding row from the first matrix and column from the second matrix.So, let's compute each element of the resulting matrix:1. **Top-left element**: Multiply the first row of the first matrix by the first column of the second matrix. [ a times a + frac{15}{34} times c = a^2 + frac{15}{34}c ]2. **Top-right element**: Multiply the first row of the first matrix by the second column of the second matrix. [ a times frac{15}{34} + frac{15}{34} times frac{25}{34} = frac{15}{34}a + frac{375}{1156} ]3. **Bottom-left element**: Multiply the second row of the first matrix by the first column of the second matrix. [ c times a + frac{25}{34} times c = ac + frac{25}{34}c ]4. **Bottom-right element**: Multiply the second row of the first matrix by the second column of the second matrix. [ c times frac{15}{34} + frac{25}{34} times frac{25}{34} = frac{15}{34}c + frac{625}{1156} ]So, putting it all together, the product (mathbf{P}^2) is:[begin{pmatrix} a^2 + frac{15}{34}c & frac{15}{34}a + frac{375}{1156} ac + frac{25}{34}c & frac{15}{34}c + frac{625}{1156} end{pmatrix}]Now, since (mathbf{P}^2 = mathbf{P}), each corresponding element of these two matrices must be equal. That gives us four equations:1. (a^2 + frac{15}{34}c = a)2. (frac{15}{34}a + frac{375}{1156} = frac{15}{34})3. (ac + frac{25}{34}c = c)4. (frac{15}{34}c + frac{625}{1156} = frac{25}{34})Hmm, that's four equations, but actually, some of them might be redundant or related. Let me see which ones are independent.Looking at equation 2 and equation 4, they both involve only one variable each, so maybe I can solve for (a) and (c) directly from those.Starting with equation 2:[frac{15}{34}a + frac{375}{1156} = frac{15}{34}]Let me subtract (frac{375}{1156}) from both sides to isolate the term with (a):[frac{15}{34}a = frac{15}{34} - frac{375}{1156}]To compute the right-hand side, I need a common denominator. The denominators are 34 and 1156. Since 34 squared is 1156, I can convert (frac{15}{34}) to have denominator 1156:[frac{15}{34} = frac{15 times 34}{34 times 34} = frac{510}{1156}]So, substituting back:[frac{15}{34}a = frac{510}{1156} - frac{375}{1156} = frac{135}{1156}]Now, solve for (a):[a = frac{135}{1156} div frac{15}{34} = frac{135}{1156} times frac{34}{15}]Simplify this:First, notice that 34 and 1156: 1156 divided by 34 is 34. So, 34 cancels out with 1156, leaving 34 in the denominator.Similarly, 135 divided by 15 is 9.So,[a = frac{9}{34}]Alright, got (a = frac{9}{34}). Now, let's solve for (c) using equation 4:[frac{15}{34}c + frac{625}{1156} = frac{25}{34}]Subtract (frac{625}{1156}) from both sides:[frac{15}{34}c = frac{25}{34} - frac{625}{1156}]Again, find a common denominator for the right-hand side. Convert (frac{25}{34}) to have denominator 1156:[frac{25}{34} = frac{25 times 34}{34 times 34} = frac{850}{1156}]So,[frac{15}{34}c = frac{850}{1156} - frac{625}{1156} = frac{225}{1156}]Now, solve for (c):[c = frac{225}{1156} div frac{15}{34} = frac{225}{1156} times frac{34}{15}]Simplify:34 cancels out with 1156, leaving 34 in the denominator.225 divided by 15 is 15.So,[c = frac{15}{34}]Alright, so I have (a = frac{9}{34}) and (c = frac{15}{34}). Let me check if these satisfy the other equations, just to be thorough.Looking at equation 1:[a^2 + frac{15}{34}c = a]Substitute (a = frac{9}{34}) and (c = frac{15}{34}):Left-hand side:[left(frac{9}{34}right)^2 + frac{15}{34} times frac{15}{34} = frac{81}{1156} + frac{225}{1156} = frac{306}{1156}]Simplify (frac{306}{1156}):Divide numerator and denominator by 2: (frac{153}{578}), which is approximately 0.2647.Right-hand side:[a = frac{9}{34} approx 0.2647]So, both sides are equal. Good.Now, equation 3:[ac + frac{25}{34}c = c]Substitute (a = frac{9}{34}) and (c = frac{15}{34}):Left-hand side:[frac{9}{34} times frac{15}{34} + frac{25}{34} times frac{15}{34} = frac{135}{1156} + frac{375}{1156} = frac{510}{1156}]Simplify (frac{510}{1156}):Divide numerator and denominator by 34: (frac{15}{34}).Right-hand side:[c = frac{15}{34}]So, both sides are equal. Perfect.Therefore, the values (a = frac{9}{34}) and (c = frac{15}{34}) satisfy all four equations, confirming that the matrix is indeed a projection matrix with these values.**Final Answer**The ordered pair is boxed{left( frac{9}{34}, frac{15}{34} right)}.

💡Alright, so I've got this problem here that I need to solve. It says, "Find the value of ( x ) satisfying the equation: ( x(x+2) + 1 = 36 )." Okay, let's break this down step by step. I'm going to take it slow because I want to make sure I understand each part correctly.First, I see that the equation is ( x(x+2) + 1 = 36 ). Hmm, that looks like a quadratic equation. Quadratic equations are of the form ( ax^2 + bx + c = 0 ), right? So, I think I need to rearrange this equation to match that standard form.Let me start by expanding the left side of the equation. So, ( x(x+2) ) would be ( x^2 + 2x ). Then, adding 1 gives me ( x^2 + 2x + 1 ). So, the equation now looks like ( x^2 + 2x + 1 = 36 ).Okay, now I need to get everything on one side to set the equation equal to zero. I'll subtract 36 from both sides:( x^2 + 2x + 1 - 36 = 0 )Simplifying that, I get:( x^2 + 2x - 35 = 0 )Alright, so now I have a quadratic equation: ( x^2 + 2x - 35 = 0 ). To solve this, I can use the quadratic formula, which is ( x = frac{-b pm sqrt{b^2 - 4ac}}{2a} ), where ( a = 1 ), ( b = 2 ), and ( c = -35 ).Plugging in those values:( x = frac{-2 pm sqrt{(2)^2 - 4(1)(-35)}}{2(1)} )Calculating inside the square root:( 2^2 = 4 )( 4 times 1 times (-35) = -140 )So, ( 4 - (-140) = 4 + 140 = 144 )Now, the square root of 144 is 12. So, the equation becomes:( x = frac{-2 pm 12}{2} )This gives me two possible solutions:1. ( x = frac{-2 + 12}{2} = frac{10}{2} = 5 )2. ( x = frac{-2 - 12}{2} = frac{-14}{2} = -7 )So, the solutions are ( x = 5 ) and ( x = -7 ).Wait a minute, let me check if these solutions make sense in the original equation. Plugging ( x = 5 ) back into ( x(x+2) + 1 ):( 5(5 + 2) + 1 = 5 times 7 + 1 = 35 + 1 = 36 )That works. Now, checking ( x = -7 ):( (-7)(-7 + 2) + 1 = (-7)(-5) + 1 = 35 + 1 = 36 )That also works. So, both solutions are valid.Hmm, but the problem asks for "the value of ( x )", which might imply a single answer. Maybe I need to consider the context or if there's a restriction on ( x ). The problem doesn't specify any restrictions, so both solutions are correct.But wait, looking back at the equation ( x(x+2) + 1 = 36 ), it's a quadratic, so it's expected to have two solutions. So, I think both ( x = 5 ) and ( x = -7 ) are correct.However, sometimes problems might expect the positive solution if it's in a real-world context, but since there's no context given here, I think both are valid.Let me double-check my calculations to be sure I didn't make any mistakes.Starting with the original equation:( x(x + 2) + 1 = 36 )Expanding:( x^2 + 2x + 1 = 36 )Subtracting 36:( x^2 + 2x - 35 = 0 )Using the quadratic formula:( x = frac{-2 pm sqrt{4 + 140}}{2} = frac{-2 pm 12}{2} )Which gives ( x = 5 ) and ( x = -7 ). Yep, that seems right.Alternatively, I could have factored the quadratic equation ( x^2 + 2x - 35 = 0 ). Let's see if that's possible.Looking for two numbers that multiply to -35 and add up to 2. Hmm, 7 and -5: 7 * (-5) = -35 and 7 + (-5) = 2. Perfect!So, the equation factors to:( (x + 7)(x - 5) = 0 )Setting each factor equal to zero:1. ( x + 7 = 0 ) ⇒ ( x = -7 )2. ( x - 5 = 0 ) ⇒ ( x = 5 )Same solutions as before. So, that confirms it.I think I've got it. The solutions are ( x = 5 ) and ( x = -7 ). Since the problem asks for "the value of ( x )", and there are two solutions, I should probably present both unless instructed otherwise.But just to be thorough, let me consider if there's any reason to discard one of the solutions. For example, if ( x ) represented a physical quantity that can't be negative, we might discard ( x = -7 ). But since there's no such context here, both solutions are valid.So, my final answer is ( x = 5 ) and ( x = -7 ).