# Find the minimum and maximum distance from a given point to a given curve.

To solve this problem I adapt the Lagrange multipliers method used here

https://math.boisestate.edu/~jaimos/classes/m275-spring2017/notes/lagrange-multipliers.html

This way my attempt to solve this problem, goes as follow:

Let $$f(x,y)=(x-8)^{2}+(y-6)^{2}$$ the squared distance about the point $$(8,6)$$ and $$g(x,y)=x^{2}+y^{2}-2x-2y-23=0.$$Then $$\nabla f=2(x-8)i + 2(y-6)j$$ and $$\nabla g = 2(x-1)i + 2(y-1)j.$$ So $$\nabla f = \lambda \nabla g$$, then $$2(x-8)i +2 (y-6)= \lambda[2(x-1)i+2(y-1)j]$$. This way I obtained $$x=\frac{-8+\lambda}{\lambda-1}$$ and $$y=\frac{-6+\lambda}{\lambda-1}$$. In order to obtain the value of $$\lambda$$ I substitute the $$x$$ and $$y$$ obtained in $$x^{2}+y^{2}-2x-2y-23=0$$. But is really hard to get $$\lambda$$ from this equation. I was wondering if there is an easier method to solve this problems or if not, I was wondering how to get the value of $$\lambda$$. And how to conclude this problem?? Thanks!

Only for this particular problem you have the curve $$g(x,y)=x^{2}+y^{2}-2x-2y-23=0$$, which is equation of a circle $$(x-1)^2+(y-1)^2 = 25$$ whose radius is $$5$$ and center at $$(1,1)$$. So you can easily calculate the distance.
The distance between the center and the point is $$\sqrt{(8-1)^2 + (6-1)^2} = \sqrt{74}$$, so the minimum distance is $$\sqrt{74}-5$$, maximum distance is $$\sqrt{74}+5$$
• Wow! Thanks a lot! Didnt see that nice factorization of $g(x,y)$. No I will practise to became better to find those tricky factorizations. Also, I was wondering If you can find a nice factorization for $g(x,y)=x^2+4x-y^2-4y=8$, I thought it was the circle equation at center $(2,-2)$ and radius 4 but the negative terms at $y$ trouble me :( @amitava
• Sorry for responding late, that is equation of a hyperbola, $(x+2)^2-(y+2)^2 = 8$ and it does not look like it has an easy solution like this. For a particular case (using calculus) you may want to see this math.stackexchange.com/questions/921924 May 4 '19 at 5:02