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) 3

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 \[1+\frac{4am}{c}=0\]

\[\Rightarrow \]   \[c=-\,4\,\,am\]

\[\Rightarrow \]   equation of chord is

\[y=m\,(x-4a)\]

which is always passing through (4a, 0)

Let point of intersection of tangents at the extremities is \[({{x}_{1}},\,\,{{y}_{1}})\] hence equation of chord of contact is

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

which is passing through (4a, 0)

\[\Rightarrow \]   \[{{x}_{1}}+4a=0\]

\[\Rightarrow \]   \[x+4a=0\] is the required locus.