EllipseHardBloom L3
Question
The point on the ellipse $x^{2} + 2y^{2} = 6$ closest to the line $x + y = 7$ :
Options
A.$(1,2)$
B.$(2,1)$
C.$(3,2)$
D.none of these
Solution
Sol.(b) Shortest distance is along common normal
Equation of normal at $P$ is $\sqrt{6}xsec\theta - \sqrt{3}ycosec\theta = 3$ slope $= \sqrt{2}tan\theta = 1$
$$\Rightarrow \ tan\theta = \frac{1}{\sqrt{2}}$$
$$\Rightarrow \ P \equiv \left( \sqrt{6}\frac{\sqrt{2}}{\sqrt{3}},\sqrt{3}\frac{1}{\sqrt{3}} \right) \equiv (2,1)$$
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