CAT 2017 Quantitative Ability

Start by finding the distance from some point on the curve to (2, 0) in terms of x Using the distance formula, we get $D=\sqrt{(x-2)^{2}+\left(\sqrt{16x^{2}+5x+16-0}\right)^{2}}$ $D=\sqrt{x^{2}-4x+4+16x^{2}+5x+16}$ $D=\sqrt{17x^{2}+x+20}$ This will end up being a messy derivative. However, since the distance D will never be negative, we can minimize $D^{2}$ instead of D and still get the same answer So, now we get $D^{2}=17x^{2}+x+20$ $\;\;\frac{dD^{2}}{dx}=34x+1$ Now we set this equal to 0 and solve for x 34x +1 = 0 $x = -\frac{1}{34}$ So, the distance is minimized at $x = -\frac{1}{34}$ and to find the minimum distance, simply evaluate D when $x = -\frac{1}{34}$ $\therefore \;\; D=\sqrt{\;\;\frac{17}{(34)^{2}}-\;\;\frac{1}{34}+20}$ $=\;\;\frac{1}{34}\sqrt{17-34+20\times(34)^{2}}$ $=\;\;\frac{1}{34}\sqrt{23103}$ ≈ 4.4 units