Question

If normals drawn to $y^2=12x$ makes an angle of $45°$ with $x-$axis, then foot of the normals is/are

• A. $(12,-12)$
• B. $(12,12)$
• C. $(3,-6)$
• D. $(3,6)$

Answer 1

Answer: (C), (D)

The correct options are
C $(3,-6)$
D $(3,6)$

$\because$ Normal makes an angle of $45°$ with $x-$axis

The slope of normal can be
$m=\pm \tan \left({45}^{\circ }\right)=\pm 1$
Let any point on the parabola be $(a{t}^2,2at)$
For the given parabola $y^2=12x$,
$a=3$
$\therefore$ The point is $(3t^2,6t)$
Now slope of the normal at this point is $-t$
Now,
$-t=\pm 1\Rightarrow t=\mp 1$

Hence, the foot of the normals are $(3,\pm 6)$