Show smaller circle lie inside bigger circle

[goldfish9776]

Homework Statement

I was asked to show the circle $x^2$ + $y^2$ +6$x$ -10$y$ +9=0 lies entirely inside circle $x^2$ + $y^2$ +4$x$ -6$y$-48=0...
For circle $x^2$ + $y^2$ +6$x$ -10$y$ +9=0
I managed to get the centre = (3,5) r=5

For circle $x^2$ + $y^2$ +4$x$ -6$y$-48=0
i gt centre = (2,3) r= sqrt 61How to proceed from here ??

Homework Equations

The Attempt at a Solution

 

[phinds]

Draw them on a piece of graph paper.

 

[RUber]

If you are looking to show it analytically, you could look for points of intersection by solving both equations for one variable and setting them equal. That seems like too much work for this problem though.
Also you could use the radius of the larger circle. Find the radius that passes through the center of the smaller circle. If the sum of the distance between the centers and the radius of the smaller circle is less than the radius of the larger, then you know it will be entirely inside the larger circle.

 

[haruspex]

If you are looking to show it analytically, you could look for points of intersection ... That seems like too much work for this problem though.

Are you sure? I think it might be the easiest way, but not by solving either in isolation. You can get a linear equation very quickly.

 

[BvU]

It doesn't matter for this exercise, but your center points are (-3,5) and (-2,3)...

Also, if you draw two circles, one inside the other (and preferably not concentric ), you quickly see that the points of closest approach are on a line through the centers. And you can comfortably see what RUber means. (Comfortably meaning: it's easier to see than to describe in words)

 

[ehild]

See picture. If you shrink both circles by the radius of the smaller one, the small circle becomes a point and you need to figure out if it is inside the shrunk big circle.