Question

The coordinates of the point of contact of the tangent to the parabola $y^2=16x$, which is perpendicular to the line $2x-y+5=0$ are

• A. (16, 16)
• B. (16, -16)
• C. (1, 4)
• D. (1, -4)

Answer 1

The correct option is B

(16, -16)

$y^2=16x.........\left(1\right)4a=16\Rightarrow a=42x-y+5=0.......\left(2\right)$
A line is perpendicular to the line (2) is
$x+2y+\lambda =0......\left(3\right)\Rightarrow Slope,m=\frac{-1}{2}$
Slope of the tangent to a parabola at $(a{t}^2,2at)$ is $\frac{1}{t}$. Hence, a line having slope m intersects the parabola at $(\frac{a}{m^2},\frac{2a}{m})$
Line (3) touches the parabola (1). So, the coordinates of the point of contact is
$\Rightarrow P(\frac{a}{m^2},\frac{2a}{m})=(16,-16)$