Question

The tangents at two points $P$ and $Q$ on the parabola $y^{2}$ $=4 x$ intersect at $T$. If $S P, S T$ and $S Q$ are equal to $a, b$ an $c$ respectively, where $S$ is the focus, then the roots of the equation $a x^{2}+2 b x+c=0$ are

• A. real and equal
• B. real and unequal
• C. complex numbers
• D. irrational

Answer 1

Answer: (A)

The tangents at the points $P\left(t_{1}^{2}, 2 t_{1}\right)$ and $Q\left(t_{2}^{2}, 2 t_{2}\right)$ intersect at the point $T\left(t_{1} t_{2}, t_{1}+t_{2}\right)$.

Now, $a=S P=1+t_{1}^{2}$ and $c=S Q=1+t_{2}^{2}$
$ \therefore b^{2} =S T^{2}=\left(t_{1} t_{2}-1\right)^{2}+\left(t_{1}+t_{2}\right)^{2}$
$=t_{1}^{2}+t_{2}^{2}+1+t_{1}^{2} t_{2}^{2}$
$=\left(1+t_{1}^{2}\right)\left(1+t_{2}^{2}\right)=a c $
$\therefore $ Roots of the equation $a x^{2}+2 b x+c=0$ are real and equal.