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

  • Thread starter Thread starter Monoxdifly
  • Start date Start date
  • Tags Tags
    Parabola
AI Thread Summary
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.
Monoxdifly
MHB
Messages
288
Reaction score
0
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.
 
Mathematics news on Phys.org
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]
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
Is it possible to arrange six pencils such that each one touches the other five? If so, how? This is an adaption of a Martin Gardner puzzle only I changed it from cigarettes to pencils and left out the clues because PF folks don’t need clues. From the book “My Best Mathematical and Logic Puzzles”. Dover, 1994.

Similar threads

Back
Top