Sliding parabola

Geometry Level pending

If a parabola whose length of latus rectum is 4a touches both the co-ordinate(tangent to axes) then locus of its focus

xy=a²(x²+y²) x²-y²=a²(x²+y²) x²+y²=4a² x²y²=a²(x²-y²) x²y²=a²(x²+y²)

1 solution

Michael Mendrin
Aug 8, 2018

We can use the parametric equations for a parabola of lactus rectum length of $4a$ , rotated by $\theta$ , with the focus at the origin $(0,0)$

$x(t)=\dfrac { 2aCos(t-\theta) } { 1+Cos(t) }$

$y(t)=\dfrac { 2a Sin(t-\theta) } { 1+Cos(t) }$

To find the distance of the horizontal and vertical tangents from the origin, we solve the following equations for $t$

$x'(t)=0$
$y'(t)=0$

so that we end up with the distances as a function of $\theta$

$X(\theta)=\dfrac{2aCos(\theta)}{1+Cos(2\theta)}$

$Y(\theta)=\dfrac{2aSin(\theta)}{1+Cos(2\theta)}$

which satisfies the equation

$(X^2+Y^2)=a^2X^2Y^2$