Question

Find the tangent to the parabola $y^2=8x$ which makes an angle of ${45}^{\circ }$ to the line
$2x+y+3=0$

• A. $x+3y+6=0$
• B. $x-3y-12=0$
• C. $9x+3y-2=0$
• D. $9x-3y-2=0$

Answer 1

The correct option is A $x+3y+6=0$

Given parabola is $y^2=8x$

Also its given that tangent makes ${45}^{\circ }$ with the line 2x+y+3=0

If m is the slpoe of tangent then,

$\left|\frac{m-(-2)}{1+m(-2)}\right|=1$

$\frac{m+2}{1-2m}=\pm 1$

$\therefore m=3orm=-\frac{1}{3}$

Now for a tangent with slope m to the parabola $y^2=4ax$.

We can use the formula.

$y=mx+\frac{a}{m}$

here $m=3or-\frac{1}{3}$

a=2

$\therefore$ Required equation are

$y=3x+\frac{2}{3}\text{or}y=-\frac{1}{3}x-\frac{2\left(3\right)}{3}$

$\Rightarrow 9x-3y+2=0\text{or}x+3y+6=0$