The circles $x^{2} + y^{2} = 400$ and $x^{2} + y^{2} - 10 x - 24 y + 80 = 0$ are in same plane . Which of the following statement is true ?

They do not touch each other.

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The intersect each other in two points.

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They touch each other externally.

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They touch each other internally.

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Solution

The correct option is D They touch each other internally.
The centres of the circles are O(0, 0) and A(5, 12), respectively.
The radii are $r_{1}$ = 20 and $r_{2}$ = 7, respectively.
OA = 13 = $r_{1} - r_{2}$ implies that the two circles touch internally.
Let P be the point of contact so that
OP:PA = 20: -7.
Therefore P = $(\frac{100}{13}, \frac{240}{13})$

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