Lagrange's Multiplier Method

Question

I need to find the distance between the ellipse $\frac{x^2}{4} + \frac{y^2}{9} = 1$ and the line $y = 10 - 2x$ using Lagranges' Multiplier Method. So far I know how to find the minimum distance between origin and any curve using this method but unable to apply the concept in this case.

Answer 1

hint:

line $ y=10-2x$ can be written as $2x+y-10=0$

for any point$(x,y)$, the distance $d$ of the point to the line $ax+by+c=0 $ is

$d=\dfrac{|ax+by+c|}{\sqrt{a^2+b^2}}=f(x,y)$

and the point is on the ellipse $\implies g=\dfrac{x^2}{4} + \dfrac{y^2}{9} - 1$

$F=f+ \lambda g$