Question

If the circles x² + y² = 9 and x² + y² + 8y + c = 0 touch each other, then c is equal to

Options

• 15

• -15

• 16

• -16

Answer 1

15

The centre of the circle x² + y² = 9 is (0, 0).
Let us denote it by C₁.
The centre of the circle x² + y²+ 8y + c = 0 is (0, −4).
Let us denote it by C₂.
The radius of x² + y² = 9 is 3 units.
   x² + y²+ 8y + c = 0

\[\Rightarrow \left( x - 0 \right)^2 + \left( y + 4 \right)^2 = 16 - c = \left( \sqrt{16 - c} \right)^2\]

Therefore, the radius of the above circle is \[\sqrt{16 - c}\].

Let the circles touch each other at P.
∴ C₁C₂ = PC₂ + PC₁
⇒ PC₂ = 4 − 3 = 1
⇒ PC₂ = 1 = \[\sqrt{16 - c}\]

⇒ c = 15