Question

A tangent to the parabola $y^2=8x$ makes an angle of ${45}^{\circ }$ with the straight line $y=3x+5$. Then one of the points of contact has the coordinates

• A. (2, 4)
• B. (2, -4)
• C. (8, -8)
• D. (8, 8)

Answer 1

The correct option is D

(8, 8)

Equation of the tangent to the parabola $y^2=4ax$ at $(a{t}^2,2at)$ is $ty=x+a{t}^2.$ Here a =2. So, the equation of the tangent at $2t^2,4t$ to the parabola
$y^2=8x$ is
$ty=x+2t^2$..............(1)
Slope of (1) is $\frac{1}{t}$ and of the given line is 3.
$\Rightarrow \frac{\frac{1}{t}-3}{1+\frac{1}{t}.3}=\pm tan{45}^o=\pm 1\Rightarrow t=\frac{-1}{2}or2$
$Fort=\frac{-1}{2},thetangentis\left(\frac{-1}{2}\right)y=x+2\left(\frac{1}{4}\right)\Rightarrow 2x+y+1=0$ at point of contact $(\frac{1}{2},-2)$
For t =2, the tangent is 2y = x+8, at point of contact (8, 8)