Single Choice

The line $$x = y$$ touches a circle at the point $$(1, 1)$$. If the circle also passes through the point $$(1, -3)$$ then its radius is:

A$$3\sqrt{2}$$
Correct Answer
B$$3$$
C$$2\sqrt{2}$$
D$$2$$

Solution

Equation of circle can be written as
$$(x - 1)^2 + (y - 1)^2 + \lambda (x - y) = 0$$
If passes through $$(1, 3)$$
$$16 + \lambda (4) = 0 \Rightarrow \lambda = -4$$
So $$(x - 1)^2 + (y - 1)^2 - 4(x - y) = 0$$
$$\Rightarrow x^2 + y^2 - 6x + 2y + 2 = 0$$
$$\Rightarrow r = 2\sqrt{2}$$

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