Question

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

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

Answer 1

Answer: (A)

The tangents at the points $P\left(t_1^2,2t_1\right)$ and $Q\left(t_2^2,2t_2\right)$ intersect at the point $T\left(t_1{t}_2,t_1+t_2\right)$.

Now, $a=SP=1+t_1^2$ and $c=SQ=1+t_2^2$
$\therefore b^2=S{T}^2={\left(t_1{t}_2-1\right)}^2+{\left(t_1+t_2\right)}^2$
$=t_1^2+t_2^2+1+t_1^2{t}_2^2$
$=\left(1+t_1^2\right)\left(1+t_2^2\right)=ac$
$\therefore$ Roots of the equation $a{x}^2+2bx+c=0$ are real and equal.