Question

If the line segment intercepted by the parabola $y^2=4\alpha x$ on the line $ax+by+c=0$ subtends a right angle at the vertex, then

• A. $4a\alpha +c=0$
• B. $4b\alpha +c=0$
• C. $4a\alpha +b=0$
• D. none of these

Answer 1

Answer: (A)

$4a\alpha +c=0$

Making the equation of parabola $y^2=4\alpha x$ homogeneous using the equation of line $ax+by+c=0$, we get

$y^2=4\alpha x\left(\frac{ax+by}{-c}\right)\Rightarrow 4a\alpha x^2+4b\alpha xy+c{y}^2=0$

which represents the combined equation of $OP$ and $OQ$.

Since $\angle POQ={90}^0$, coefficients of $x^2+$ coefficients of $y^2=0$

$\Rightarrow 4a\alpha +c=0$