First slide

Analysis of two circles in circles

Question

The equation of the circle passing through (0, 0) and belonging to the system of circles of which (3, 1) and (-1, 5) are limiting points, is

Moderate

A

$x^{2} + y^{2} - x + 3 y = 0$
B

$x^{2} + y^{2} - 11 x + 3 y = 0$
C

$x^{2} + y^{2} = 1$
D

none of these

Solution

Circles having( 3, 1) and (-1, 5) as limiting points
are $S_{1} \equiv (x - 3)^{2} + (y - 1)^{2} = 0$ and $S_{2} \equiv (x + 1)^{2} + (y - 5)^{2} = 0$
The equation of the family of circles is
$\begin{array}{l} S_{1} + λ (S_{1} - S_{2}) = 0 \\ \Rightarrow (x - 3)^{2} + (y - 1)^{2} + λ (- 8 x + 8 y - 16) = 0 \end{array}$ ... (i)
It passes through (0, 0).
$∴ 10 - 16 λ = 0 \Rightarrow λ = \frac{5}{8}$
Substituting the value of $λ$ in (i), we get
$x^{2} + y^{2} - 11 x + 3 y = 0$ as the equation of the required circle.

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