Question

The point at shortest distance from the line \[x+y=7\] and lying on an ellipse \[x^{2}+2y^{2}=6,\] has coordinates______.

विकल्प

• \[(\sqrt2,\sqrt2)\]

• \[(0,\sqrt3)\]

• (2,1)

• \[\left(\sqrt{5},\frac{1}{\sqrt{2}}\right)\]

Answer 1

The point at shortest distance from the line \[x+y=7\] and lying on an ellipse \[x^{2}+2y^{2}=6,\] has coordinates (2,1).

Explanation:

The tangent at the point of shortest distance from the line \[x+y=7\] parallel to the given line.

Any point on the given ellipse is \[(\sqrt{6}\cos\theta,\sqrt{3}\sin\theta).\]

Equation of the tangent is \[{\frac{x\cos\theta}{\sqrt{6}}}+{\frac{y\sin\theta}{\sqrt{3}}}=1\] which is parallel to \[x+y=7\].

\[\Rightarrow\quad\frac{\cos\theta}{\sqrt{6}}=\frac{\sin\theta}{\sqrt{3}}\]

\[\Rightarrow\quad\frac{\cos\theta}{\sqrt{2}}=\frac{\sin\theta}{1}=\frac{1}{\sqrt{3}}\]

\[\therefore\] The required point is (2, 1).