MHB How to Determine the Reflection of a Parabola by a Given Line?

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To determine the reflection of the parabola defined by the equation y^2 - 2y - 4x - 11 = 0 across the line y = -x, the coordinates of any point (x, y) transform to (-y, -x). By substituting these transformed coordinates into the original parabola equation, the reflection can be derived. The resulting equation after substitution is x^2 + 2x + 4y - 11 = 0. This algebraic method provides a clear approach to finding the reflected parabola. The discussion emphasizes the importance of understanding point transformation in reflections.
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Determine the reflection of a parabola $$y^2-2y-4x-11=0$$ by the line y = -x.

I know how to do it graphically, but please tell me how to do it algebraically.
 
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Monoxdifly said:
Determine the reflection of a parabola $$y^2-2y-4x-11=0$$ by the line y = -x.

I know how to do it graphically, but please tell me how to do it algebraically.
In such a reflection, the images of the points (1,0) and (0,1) are (0, -1) and (-1,0), respectively.

This means that the image of (x,y) is (-y, -x). You only need to substitute that in the equation.
 
$$(-x)^2-2(-x)-4(-y)-11=0$$?
 
Monoxdifly said:
$$(-x)^2-2(-x)-4(-y)-11=0$$?
Yes
 
[DESMOS=-20,20,-13.35559265442404,13.35559265442404]y^2-2y-4x-11=0;x^2+2x+4y-11=0;y=-x;[/DESMOS]
 

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