question_answer

A variable chord PQ of parabola \[{{y}^{2}}=4ax\] subtends a right angle at the vertex. Find the locus of point of intersection of the tangents at P and Q.

A) 0

B) 1    

C) 2         

D) none

Correct Answer: A

Solution :

Let \[y=mx+c\] is the variable chord which is subtending right angle at the vertex of parabola hence equation of pair of straight lines OP and OQ can be given by making a homogeneous second degree equation with the help of parabola and chord as follows

\[{{y}^{2}}-4ax\left( \frac{y-mx}{c} \right)=0\]

for subtend \[90{}^\circ \] at vertex \[1+\frac{4am}{c}=0\]

\[\Rightarrow c=-\,4am\]\[\Rightarrow \]equation of chord is \[y=m\,\,(x-4a)\]

Let point of intersection of tangents at the extremeities is \[({{x}_{1}},{{y}_{1}})\]

\[y\,{{y}_{1}}=2a(x+{{x}_{1}})\]

Which is passing through \[(4a,0)\]

\[\Rightarrow {{x}_{1}}+4a=0\]\[\Rightarrow x+4a=0\] is the required locus.