Question

Find the coordinates of a point on the parabola y=x²+7x + 2 which is closest to the strainght line y = 3x $-$ 3.

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{coordinates}\mathrm{of}\mathrm{the}\mathrm{point}\mathrm{on}\mathrm{the}\mathrm{parabola}\mathrm{be}\left(x,y\right).\mathrm{Then}, \\ y=x^2+7x+2...\left(1\right) \\ \mathrm{Let}\mathrm{the}\mathrm{distance}\mathrm{of}\mathrm{a}\mathrm{point}\left(x,\left(x^2+7x+2\right)\right)\mathrm{from}\mathrm{the}\mathrm{line}y=3x-3\mathrm{be}S.\mathrm{Then}, \\ S=\left|\frac{-3x+\left(x^2+7x+2\right)+3}{\sqrt{10}}\right| \\ \Rightarrow \frac{dS}{dt}=\frac{-3+2x+7}{\sqrt{10}} \\ \mathrm{For}\mathrm{maximum}\mathrm{or}\mathrm{minimum}\mathrm{values}\mathrm{of}S,\mathrm{we}\mathrm{must}\mathrm{have} \\ \frac{dS}{dt}=0 \\ \Rightarrow \frac{-3+2x+7}{\sqrt{10}}=0 \\ \Rightarrow 2x=-4 \\ \Rightarrow x=-2 \\ \mathrm{Now}, \\ \frac{d^{2}S}{d{t}^2}=\frac{2}{\sqrt{10}}>0 \\ \mathrm{So},\mathrm{the}\mathrm{nearest}\mathrm{point}\mathrm{is}\left(x,\left(x^2+7x+2\right)\right). \\ \Rightarrow \left(-2,4-14+2\right) \\ \Rightarrow \left(-2,-8\right) \end{gathered}$$