Find the condition that the straight line $A x + B y + C = 0$ may touch the circle.
$(x - a)^{2} + (y - b)^{2} = c^{2}$ .

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Solution

Fact: If a line is tangent to a circle, then distance of centre of the circle from the line will be same as its radius
Here the center of the circle is $(a, b)$ , and radius is $c$
$\Rightarrow ∣ ∣ ∣ \frac{A (a) + B (b) + C}{\sqrt{A^{2} + B^{2}}} ∣ ∣ ∣ = c$
$∴ A a + B b + C = \pm c \sqrt{A^{2} + B^{2}}$

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