Answer:
-72x - 53y + 287 = 0.
Explanation:
To find the equation of the image of the circle, we need to reflect each point on the circle in the given line mirror.
The line mirror equation is given as 4x + 7y + 13 = 0.
The reflection of a point (x, y) in the line mirror can be found using the formula:
x' = (x - 2Ay - 2B(Ax + By + C)) / (A^2 + B^2)
y' = (y - 2Bx + 2A(Ax + By + C)) / (A^2 + B^2)
where A, B, and C are the coefficients of the line mirror equation.
For the given line mirror equation 4x + 7y + 13 = 0, we have A = 4, B = 7, and C = 13.
Now, let's find the equations of the image of the circle.
The original circle equation is x² + y² + 16x - 24y + 183 = 0.
Using the reflection formulas, we substitute the values of x and y in the circle equation to find x' and y':
x' = (x - 2Ay - 2B(Ax + By + C)) / (A^2 + B^2)
= (x - 2(4)y - 2(7)(4x + 7y + 13)) / (4^2 + 7^2)
= (x - 8y - 8(4x + 7y + 13)) / 65
= (x - 8y - 32x - 56y - 104) / 65
= (-31x - 64y - 104) / 65
y' = (y - 2Bx + 2A(Ax + By + C)) / (A^2 + B^2)
= (y - 2(7)x + 2(4)(Ax + By + C)) / (4^2 + 7^2)
= (y - 14x + 8(Ax + By + C)) / 65
= (y - 14x + 8(4x + 7y + 13)) / 65
= (57x + 35y + 104) / 65
Therefore, the equation of the image of the circle is:
(-31x - 64y - 104) / 65 + (-57x + 35y + 104) / 65 + 16x - 24y + 183 = 0
Simplifying the equation, we get:
-31x - 64y - 57x + 35y + 16x - 24y + 183 + 104 = 0
-72x - 53y + 287 = 0
So, the equation of the image of the circle is -72x - 53y + 287 = 0.