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Position of a Line with Respect to Circle

Find the valu...

Question

Find the value of c if the line y = 3x + c is a tangent to the circle $x^{2} + y^{2} = 10$

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Solution

The correct option is B
-10

If L is a tangent OP, the perpendicular distance of centre O from L will be equal to radius.

$\Rightarrow O P = ∣ ∣ ∣ \frac{c}{\sqrt{1^{2} + 3^{2}}} ∣ ∣ ∣ = ∣ ∣ ∣ \frac{c}{\sqrt{10}} ∣ ∣ ∣$

We have $O P = r = \sqrt{10}$

$\Rightarrow ∣ ∣ ∣ \frac{c}{\sqrt{10}} ∣ ∣ ∣ = \sqrt{10}$

$\Rightarrow \frac{c}{\sqrt{10}} = \pm \sqrt{10}$

$\Rightarrow c = \pm 10$

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