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The point on the ellipse \mathrm{x^2+2 y^2=6} whose distance from the line \mathrm{x+y=7} is minimum is \mathrm{(m, n).} Then \mathrm{m+2 n} is ___________.

Option: 1

4


Option: 2

2


Option: 3

5


Option: 4

3


Answers (1)

best_answer

The distance of a point from the line will be minimum if tangent at that point is parallel to the line. Let the point be \mathrm{\left ( x_{1},y_{1} \right )}

Tangent at \mathrm{\left(x_1, y_1\right)\, \, is \, \, x_1 x+2 y_1 y=6}

\mathrm{ \begin{aligned} & \text { Slope }=-\frac{x_1}{2 y_1}=-1=\text { slope of } x+y=7 \\\\ & \therefore \quad x_1=2 y_1 \\\\ & x_1^2+2 y_1^2=6 \\\\ & \Rightarrow 4 y_1^2+2 y_1^2=6, y_1=1, x_1=2 \end{aligned} }
The required point is (2,1).

Posted by

Divya Prakash Singh

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