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

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$2 x+y-1=0$

$x+2 y-1=0$

$2 x+y+1=0$

$x-2 y-1=0$

Solution:

The tangent to parabola $y^{2}=8 x$ is $y=m x+\frac{2}{m}$.
If this makes an angle $45^{\circ}$ with $y=3 x+5$
$\therefore \tan 45^{\circ}=\left|\frac{ m -3}{1+3 m }\right|$
$\therefore \pm 1=\frac{m-3}{1+3 m}$
$ \Rightarrow 1+3 m=m-3$ or $3-m \Rightarrow m=-2$
$\therefore$ Equation of tangent is $y=-2 x-1$
i. e. $2 x+y+1=0$ or $y=\frac{1}{2} x+4$
i. e., $x-2 y+8=0$

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