351.

The longest distance of the point (a, 0) from the curve 2x² + y² = 2x is

• 1 + a

• $\left|1 - \mathrm{a}\right|$

• $\sqrt{1 - 2\mathrm{a} +2{\mathrm{a}}^2}$

• $\sqrt{1 - 2\mathrm{a} + 3{\mathrm{a}}^2}$

$$\begin{gathered} \mathrm{Given} \mathrm{curve} \mathrm{is} \\ 2{\mathrm{x}}^2 + {\mathrm{y}}^2 = 2\mathrm{x} \\ 2{\mathrm{x}}^2 - 2\mathrm{x} + {\mathrm{y}}^2 = 0 \\ \Rightarrow 2{\left(\mathrm{x} - \frac{1}{2}\right)}^2 +\hspace{0.17em}{\mathrm{y}}^2 = \frac{1}{2} \\ \Rightarrow \frac{{\left(\mathrm{x} - \frac{1}{2}\right)}^2 }{{\displaystyle \frac{1}{4}}} + \frac{{\displaystyle \frac{{\mathrm{y}}^2}{1}}}{2} = 1 \\ \mathrm{which} \mathrm{represents} \mathrm{an} \mathrm{ellipse} \\ \mathrm{Here}, \mathrm{a} = \frac{1}{2}, \mathrm{b} = \frac{1}{\sqrt{2}}, \mathrm{h} = \frac{1}{2}, \mathrm{k} = 0 \\ \mathrm{Consider} \mathrm{a} \mathrm{point} \mathrm{P}(\mathrm{h} + \mathrm{acos}\left(\mathrm{\theta }\right), \mathrm{k} + \mathrm{bsin}\left(\mathrm{\theta }\right)) \\ = \mathrm{P}\left(\frac{1}{2} + \frac{1}{2}\cos \left(\mathrm{\theta }\right), \frac{1}{\sqrt{2}}\sin \left(\mathrm{\theta }\right)\right) \mathrm{on} \mathrm{the} \mathrm{ellipse} \mathrm{from} \mathrm{which} \mathrm{the} \mathrm{distance} \\ \mathrm{of} \mathrm{point} (\mathrm{a}, 0) \mathrm{is} \mathrm{maximum} \\ \mathrm{let} \mathrm{Q}(\mathrm{a}, 0) \\ \mathrm{Now}, \\ \mathrm{PQ} = \sqrt{{\left(\frac{1}{2} + \frac{1}{2}\cos \left(\mathrm{\theta }\right) - \mathrm{a}\right)}^2 + {\left(\frac{1}{\sqrt{2}}\sin \left(\mathrm{\theta }\right) - 0\right)}^2 } \\ = \sqrt{\frac{1}{4} + \frac{1}{4}{\cos }^2\left(\mathrm{\theta }\right) + {\mathrm{a}}^2 + \frac{1}{2}\cos \left(\mathrm{\theta }\right) - \mathrm{acos}\left(\mathrm{\theta }\right) - \mathrm{a} + \frac{1}{2}{\sin }^2\left(\mathrm{\theta }\right)} \\ \mathrm{PQ} = \sqrt{\frac{1}{2} + {\mathrm{a}}^2 - \mathrm{a} + \left(\frac{1}{2} - \mathrm{a}\right)\cos \left(\mathrm{\theta }\right) + \frac{1}{4}{\sin }^2\left(\mathrm{\theta }\right)} \\ \mathrm{Let} \mathrm{y} = {\mathrm{PQ}}^2 = \frac{1}{2} + {\mathrm{a}}^2 - \mathrm{a} + \left(\frac{1}{2} - \mathrm{a}\right)\cos \left(\mathrm{\theta }\right) + \frac{1}{4}{\sin }^2\left(\mathrm{\theta }\right) \end{gathered}$$

$$\begin{gathered} \mathrm{For} \mathrm{maxima} \mathrm{and} \mathrm{minima}, \mathrm{put} \frac{\mathrm{dy}}{\mathrm{d\theta }} = 0 \\ \Rightarrow - \left(\frac{1}{2} - \mathrm{a}\right)\sin \left(\mathrm{\theta }\right) + \frac{1}{4} . 2\sin \left(\mathrm{\theta }\right)\cos \left(\mathrm{\theta }\right) = 0 \\ \Rightarrow \sin \left(\mathrm{\theta }\right)\left(- \frac{1}{2} + \mathrm{a} + \frac{1}{2}\cos \left(\mathrm{\theta }\right)\right) = 0 \\ \Rightarrow \sin \left(\mathrm{\theta }\right) = 0 \mathrm{or} - \frac{1}{2} + \mathrm{a} + \frac{1}{2}\cos \left(\mathrm{\theta }\right) = 0 \\ \Rightarrow \mathrm{\theta } = 0 \mathrm{or} \cos \left(\mathrm{\theta }\right) = 1 - 2\mathrm{a} \\ \Rightarrow {\sin }^2\left(\mathrm{\theta }\right) = 1 - {\cos }^2\left(\mathrm{\theta }\right) \\ = 1 - {\left(1 - 2\mathrm{a}\right)}^2 \\ = - 4{\mathrm{a}}^2 + 4\mathrm{a} \end{gathered}$$

$$\begin{gathered} \mathrm{Now}, \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{d\theta }}^2} < 0 \mathrm{for} \cos \left(\mathrm{\theta }\right) = 1 - 2\mathrm{a} \\ \mathrm{Thus}, \mathrm{distance} \mathrm{PQ} \mathrm{is} \mathrm{maximum}, \mathrm{when} \\ \cos \left(\mathrm{\theta }\right) = 1 - 2\mathrm{a} \mathrm{and} {\sin }^2\left(\mathrm{\theta }\right) = - 4{\mathrm{a}}^2 + 4\mathrm{a} \\ \mathrm{Now}, \mathrm{required} \mathrm{longest} \mathrm{distance} \mathrm{is} \\ =\sqrt{\frac{1}{2} + {\mathrm{a}}^2 - \mathrm{a} + \left(\frac{1}{2} - \mathrm{a}\right)\left(1 - 2\mathrm{a}\right) + \frac{1}{4}\left(- 4{\mathrm{a}}^2 + 4\mathrm{a}\right)} \\ = \sqrt{\frac{1}{2} + {\mathrm{a}}^2 - \mathrm{a} + \frac{1}{2} - \mathrm{a} - \mathrm{a} + 2{\mathrm{a}}^2 - {\mathrm{a}}^2 + \mathrm{a}} \\ = \sqrt{2{\mathrm{a}}^2 - 2\mathrm{a} + 1} \\ = \sqrt{1 - 2\mathrm{a} + 2{\mathrm{a}}^2} \end{gathered}$$