Question

The tangents from the origin to the parabola $y^2+4=4x$ are inclined at

• A. $\frac{\pi }{6}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{3}$
• D. $\frac{\pi }{2}$

Answer 1

Answer: (D)

$\frac{\pi }{2}$

Given equation of parabola is
$y^2+4=4x$ ....(1)
Let the equation of tangent to parabola be
$y=mx$ ....(2) (Since, passing through origin so c=0)
Let $P(x_1,y_1)$ be the point of contact.
$\Rightarrow (m{x}_1{)}^2+4=4x$
$\Rightarrow m^2{x}_1^2-4x+4=0$ ....(3)
Since,$y=mx$ is tangent . So discriminant of eqn (3) must be $0$
$\Rightarrow 16-16m^2=0$
$\Rightarrow m=\pm 1$
So, equation of tangents are $y=x$ and $y=-x$
Clearly the lines are perpendicular. ($m_1{m}_2=-1$)