Question

Show that if tangents be drawn from any point on the line $x+4a=0$, to the parabola $y^2=4ax$. Prove that their chord of contact will subtend a right at the vertex.

Answer 1

Solution:-

Equation to the chord of contact of the tangents drawn from a point $(x_1,y_1)$, to the parabola $y^2=4ax$ is-

$y{y}_1=2a(x+x_1)$

Parameterize parabola as $(at²,2at)$, so tangent at t is-

$2aty=2a(x+a{t}^2)$

$\Rightarrow ty=x+a{t}^2$

For points $P\left(p\right)\&Q\left(q\right)$, tangents are-

$py=x+a{p}^2$

$\Rightarrow x=py-a{p}^2⟶\left(i\right)$

$qy=x+a{q}^2$

$\Rightarrow x=qy-a{q}^2⟶\left(ii\right)$

From ${eq}^n\left(i\right)\&\left(ii\right)$, we have

$py-a{p}^2=qy-a{q}^2$

$\Rightarrow y(p-q)=a(p^2-q^2)$

$\Rightarrow y=a(p+q)\{\because a^2-b^2=(a-b)(a+b)\}$

Substituting value of y in ${eq}^n\left(i\right)$, we have

$x=p\times \left(a(p+q)\right)-a{p}^2$

$\Rightarrow x=apq$

If the point $(apq,a(p+q))$ lies on $x+4a=0$,

$\therefore apq+4a=0$

$\Rightarrow pq=-4⟶\left(iii\right)$

But if $OP\perp OQ$ then $\frac{2ap}{\left(a{p}^2\right)}\times \frac{2aq}{\left(a{q}^2\right)}=-1\to pq=-4$ which is true because of ${eq}^n\left(iii\right)$.

$\therefore \angle POQ=90°$

Hence, the chord of contact will subtend a right at the vertex.