First slide

Definition of a circle

Question

Equation of a circle with centre $(- 4, 3)$ touching internally and containing the circle $x^{2} + y^{2} = 1$ is

Moderate

A

$x^{2} + y^{2} + 8 x - 6 y + 9 = 0$
B

$x^{2} + y^{2} - 8 x + 6 y + 9 = 0$
C

$x^{2} + y^{2} + 8 x - 6 y - 11 = 0$
D

$x^{2} + y^{2} - 8 x + 6 y - 11 = 0$

Solution

Let the equation of the required circle be.
$(x + 4)^{2} + (y - 3)^{2} = r^{2}$ (i)
If (i) touches the circle $x^{2} + y^{2} = 1$ internally (ii)
the distance between the centres $(- 4, 3)$ and $(0, 0)$ of these circles is equal to the difference of their radii
$\Rightarrow \sqrt{4^{2} + 3^{2}} = r - 1 \Rightarrow r = 5 + 1 \Rightarrow r = 6$
So that an equation of the required circle is
$x^{2} + y^{2} + 8 x - 6 y - 11 = 0 .$

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