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  • A variable chord PQ of the parabola  substends a right angle at the vertex.

A variable chord PQ of the parabola y=4 x^2 substends a right angle at the vertex. Then the locus of points of intersection of the tangents at P and Q is 

 

Option: 1

4 y+1=16 x^2


Option: 2

y+4=0


Option: 3

4 y+4=4 x^2


Option: 4

4 y+1=0


Answers (1)

best_answer

\text { Let } P\left(t_1, 4 t_1^2\right), Q\left(t_2, 4 t_2^2\right)

Slope of OP x slope of OQ = -1

\Rightarrow 4 t_1 \cdot 4 t_2=-1

Eq of tangent at \left(t_1, 4 t_1^2\right) i s

y-4 t_1^2=8 t_1\left(x-t_1\right) \Rightarrow y+4 t_1^2=8 t_1 x

Eq of tangent at \left(t_2, 4 t_2^2\right) \text { is } y+4 t_2^2=8 t_2 x

Let (x1 , y1) is the point of intersection

\begin{aligned} & e q(1)-e q(2) \Rightarrow x_1=\frac{t_1+t_2}{2} \\ & y_1=8 t_1\left(\frac{t_1+t_2}{2}\right)-4 t_1^2=4 t_1 t_2=\frac{-1}{4} \\ & \Rightarrow 4 y_1+1=0 \end{aligned}

 

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