Find the minimum distance between the points on the ellipse
$\ \frac{x^2}{4} + y^2 = 1 $ and the straight line $\ x+y = 4$,
I know one way is to use Lagrange multiplier, let $\ L(x_1,x_2,y_1,y_2,\lambda_1, \lambda_2) = (x_1 -x_2)^2 + (y_1-y_2)^2 + \lambda_1(\frac{x_1^2}{4} + y_1^2 -1) + \lambda_2(x_2+y_2 -4)= 0 $
and differentiate with respect to each of the component, but when solving these equations, it gets very complicated…
Does anyone has some quicker ways for solving this this problem and this type of problems?
Best Answer
Hint...any point on the ellipse can be written as $$(2\cos\theta,\sin\theta)$$
The distance from this point to the line is $$\left|\frac{2\cos\theta+\sin\theta-4}{\sqrt{2}}\right|$$