question_answer

The straight line joining any point \[P\] on the parabola \[{{y}^{2}}=4ax\] to the vertex and perpendicular from the focus to the tangent at \[P\], intersect at \[R\], then the equation of the locus of \[R\] is-

A) \[{{x}^{2}}+2{{y}^{2}}-ax=0\]

B) \[2{{x}^{2}}+{{y}^{2}}-2ax=0\]

C) \[2{{x}^{2}}+2{{y}^{2}}-ay=0\]          

D) \[2{{x}^{2}}+{{y}^{2}}-2ay=0\]

Correct Answer: B

Solution :

            \[T:ty=x+a{{t}^{2}}\]                                 ? (1) Line perpendicular to \[(1)\] through \[(a,\,\,0)\]             \[tx+y=ta\]                              ? (2) Equation of\[OP:\]\[y-\frac{2}{t}x=0\]                       ? (3) From equations (2) and (3) eliminating \[t\] we get locus