Question

If the normals at points $P$ and $Q$ intersect on the parabola itself, then ordinates of $P$ and $Q$ are the roots of $y^2+ky+w{a}^2=0$ where $k$ is ordinate of the point of intersection of the normals. Find the value of $w$.

• A. $w=2$
• B. $w=4$
• C. $w=6$
• D. $w=8$

Answer 1

Answer: (D)

$w=8$

If the normals at $t_1$ and $t_2$ to the parabola meet on the parabola,

So, $t_1{t}_2=2.$ Ordinates are $2a{t}_1$ and $2a{t}_2.$

If $S=sum$ and P=product then $S=2a(t_1+t_2),$
$P=2a{t}_1.2a{t}_2$
But it is given that $k=$ ordinate of point of
intersection of normals $=-a{t}_1{t}_2(t_1+t_2)$
or $k=-2a(t_1+t_2)\because t_1{t}_2=2$
$\therefore S=-k,P=4a^2.2=8a^2$
$\therefore y^2-Sy+P=0$
or $y^2+ky+8a^2=0$