Question

If $P$ is a point on the parabola $y=x^{2}+4$ which is closest to the straight line $y =4 x -1$, then the co-ordinates of $P$ are

• A. (3,13)
• B. (1,5)
• C. (-2,8)
• D. (2,8)

Answer 1

Answer: (D)

(2,8)

Answer 2

$P: y=x^{2}+4$ $k=h^{2}+4$ $L : y=4 x-1$ $y-4 x+1=0$ $d=A B=\left|\frac{k-4 h+1}{\sqrt{5}}\right|$ $=\left|\frac{h^{2}-4-4 h+1}{\sqrt{5}}\right|$ $\frac{d(d)}{d h}=\frac{2 h-4}{\sqrt{5}}=0$ $h=2$ $\frac{d^{2}(d)}{d h^{2}}=\frac{2}{\sqrt{5}}>0$ $\therefore k=4+4=8$ $\therefore$ Point (2,8)