Question

The tangent at two points $P$ and $Q$ on the parabola $y^2=4x$ intersect at $T$. If $SP,ST$ and $SQ$ are equal to $a,b$ and $c$ 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 number
• D. irrational

Answer 1

The correct option is A real and equal

Comparing with $y^2=4ax$ and given equation $y^2=4x$ we get a=1.

Let points on parabola be $P({t_1}^2,2t_1),Q({t_2}^2,2t_2)$

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

Now, $a=SP=1+{t_1}^2$ and $c=SQ=1+{t_2}^2$

$\therefore b^2={ST}^2={(t_1{t}_2-1)}^2+{(t_1+t_2)}^2={t_1}^2+{t_2}^2+1+{t_1}^2{t_2}^2=(1+{t_1}^2)(1+{t_2}^2)=ac$

$b^2-ac=0$

$4b^2-4ac=0$

$\therefore$ Roots of the equation $a{x}^2+2bx+c=0$ are real and equal