First slide

Definition of a circle

Question

The equation of the circle which has a tangent $2 x - y - 1 = 0$ at (3,5) on it and with the centre on $x + y = 5$ is

Moderate

A

$x^{2} + y^{2} + 6 x - 16 y + 28 = 0$
B

$x^{2} + y^{2} - 6 x + 16 y - 28 = 0$
C

$x^{2} + y^{2} + 6 x + 6 y - 28 = 0$
D

$x^{2} + y^{2} - 6 x - 6 y - 28 = 0$

Solution

The equation of the family of circles touching $2 x - y - 1 = 0$ at (3,5) is
$(x - 3)^{2} + (y - 5)^{2} + λ (2 x - y - 1) = 0$ ... (1)
It has its centre $(- λ + 3, \frac{λ + 10}{2})$ on the line $x + y = 5$
$∴ - λ + 3 + \frac{λ + 10}{2} = 5 \Rightarrow λ = 6$
Putting $λ = 6$ in (i), we get
$x^{2} + y^{2} + 6 x - 16 y + 28 = 0$
as the equation of the required circle.

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