Question

Equation of a tangent to the parabola $y^2=12x$ which make an angle of ${45}^0$ with line $y=3x+77$ is

• A. $2x-4y+3=0$
• B. $x-2y+12=0$
• C. $4x+2y+3=0$
• D. $2x+y-12=0$

Answer 1

Answer: (B), (C)

The correct options are
B $x-2y+12=0$
C $4x+2y+3=0$

The slope of the given line is $m=3$. There can be two slopes of lines inclined at an angle of $\frac{\pi }{4}$ with the given line. (Clockwise or anti-clockwise).
Let $m_1,m_2$ be the slopes of the required tangents.
$m_1=\frac{3-1}{1+3}=\frac{1}{2}$
$m_2=\frac{3+1}{1-3}=-2$
The equation of a tangent to the parabola $y^2=12x$ is given by $y=mx+\frac{3}{m}$
Hence, the equations of the tangents are $y=-2x-\frac{3}{2}$ and $y=\frac{1}{2}x+6$
The equations can be rewritten as $4x+2y+3=0$ and $x-2y+12=0$
Hence, options B and C are correct.