A fixed parabola y2=4 x touches a variable parabola. If the equation to the locus of the vertex of the variable parabola is another parabola P, then find the length of latus rectum of P. Note: Assume that the two parabolas are equal and the axis of the variable parabola remains parallel to the x-axis.

Question

A fixed parabola y2=4 x touches a variable parabola. If the equation to the locus of the vertex of the variable parabola is another parabola P, then find the length of latus rectum of P. Note: Assume that the two parabolas are equal and the axis of the variable parabola remains parallel to the x-axis. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Let the equation of the parabola with vertex

(h, k)

(y - k)^{2} = - l (x - h)

substituting x = \frac{y ^{2}}{l}; l = 4 a

(y - k)^{2} = - l (\frac{y ^{2}}{l} - h)

y^{2} - 2 k y + k^{2} = - y^{2} + l h

2 y^{2} - 2 k y + k^{2} - l h = 0

put D = 0

4 k^{2} - 8 (k^{2} - l h)^{2} = 0

4 k^{2} = 8 l x \Rightarrow y^{2} = 2 l x = 8 a x = 8 x \Rightarrow latus rectum=8

Let the equation of the parabola with vertex (h,k) (y−k)2=−l(x−h) substituting x=ly2​;l=4a (y−k)2=−l(ly2​−h) y2−2ky+k2=−y2+lh 2y2−2ky+k2−lh=0 put D=0 4k2−8(k2−lh)2=0 4k2=8lx⇒y2=2lx=8ax=8x⇒ latus rectum=8