First slide

Analysis of two circles in circles

Question

The circles $x^{2} + y^{2} - 10 x + 16 = 0$ and $x^{2} + y^{2} = r^{2}$ intersect each other in two distinct points if

Moderate

A

$r < 2$
B

$r < 8$
C

$2 < r < 8$
D

$2 \leq r \leq 8$

Solution

Centres of the given circles are $C_{1} (5, 0)$ and $C_{2} (0, 0)$ and their radii are $r_{1} = 3$ and $r_{2} = r$ respectively. For the circles to intersect at two distinct points, we must have
$\begin{array}{l} |r_{1} - r_{2}| < C_{1} C_{2} < r_{1} + r_{2} \\ \Rightarrow r - 3 < 5 < r + 3 \\ \Rightarrow r < 8 and r > 2 \Rightarrow 2 < r < 8 . \end{array}$

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