Question

The equation of a tangent to the parabola $y^2=8x$ which makes an angle ${45}^{\circ }$ with the line $y=3x+5$ is

• A. $2x+y+1=0$
• B. $y=2x+1$
• C. $x+2y+8=0$
• D. $x-2y+8=0$

Answer 1

Answer: (A), (D)

The correct options are
A $2x+y+1=0$
D $x-2y+8=0$

Equation of tangent in term of slope of
$y^2=8x$ is
$y=mx+\frac{2}{m}\cdots \left(i\right)$
$\because$ Angle between $\left(i\right)$ and $y=3x+5$ is ${45}^{\circ }$, then
$\Rightarrow \left|\frac{m-3}{1+3m}\right|=\tan {45}^{\circ }=1\Rightarrow \pm (m-3)=1+3m$

When $m-3=1+3m$
$\therefore m=-2$
When $-m+3=1+3m$
$\therefore m=\frac{1}{2}$

Now, the equation of tangents are
$y=-2x-1\text{and}y=\frac{x}{2}+4\Rightarrow 2x+y+1=0\text{and}x-2y+8=0$