Find the equation of the circle of radius $5$ units which lies within the

Question

Find the equation of the circle of radius $5$ units which lies within the circle $x^2 + y^2 + 14x + 10y - 26 = 0$ and which touches the given circle at the point $(-1,3)$.

Please help me.

Thanks in Advance.

Answer 1

Hints:

$$x^2+y^2+14x+10y-26=0\iff (x+7)^2+(y+5)^2=100$$

The line through $\;(-1,3)\;$ and the big circle's center passes through the little circle's center (why?), so it is on the line

$$y-2=\frac86(x+1)$$

Finally, you need the point on the above line at a distance of $\;5\;$ from $\;(-7,-5)\;$ and between $\;(-7,-5)\;$ and $\;(-1,3)\;$ .

Answer 2

Hint:

note that the center of the given circle is $C=(-7,-5)$ and the radius is $r=10$, so the center of the internal circle is the midpoint from $(-7,-5)$ and $(-1,3)$.