First slide

Definition of a circle

Question

Equations of the common tangents of the circles $x^{2} + y^{2} - 2 x - 6 y + 9 = 0$ and $x^{2} + y^{2} + 6 x - 2 y + 1 = 0$ are
A) x = 0 B) y = 4 C) 3x +4 y =10

Moderate

A

A and B
B

B and C
C

A and C
D

A, B and C

Solution

Equations of the given circles can be written as $C_{1} : (x - 1) 2 + (y - 3) 2 = 1$ having centre at $(1, 3)$ and radius $1$ and $C_{2} : (x + 3) 2 + (y - 1) 2 = 9$ having centre at $(- 3, 1)$ and radius $3$ .
Both the circles clearly touch $x = 0$ and $y = 4$

Now $|\frac{3 \times 1 + 4 \times 3 - 10}{\sqrt{3^{2} + 4^{2}}}| = 1$
and $|\frac{3 (- 3) + 4 (1) - 10}{\sqrt{3^{2} + 4^{2}}}| = 3$

Shows that both the circles touch $3 x + 4 y = 10$ also.

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