Definition of a circle

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Question

If $3 x + y = 0$ is a tangent to the circle with centre at the point (2, -1), then the equation of the other tangent to the circle from the origin, is

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Solution

It is given that $3 x + y = 0$ is a tangent to the circle with centre at $(2, - 1) .$
$∴$ Radius = Length of the 1. from $(2, - 1) o n 3 x + y = 0$
$\Rightarrow$ Radius $= \frac{6 - 1}{\sqrt{9 + 1}} = \frac{5}{\sqrt{10}} = \sqrt{\frac{5}{2}}$
So, the equation of the circle is
$(x - 2)^{2} + (y + 1)^{2} = \frac{5}{2} \Rightarrow x^{2} + y^{2} - 4 x + 2 y + \frac{5}{2} = 0$
The combined equation of the tangents drawn from the origin to this circle is
$\begin{aligned} (x^{2} + y^{2} - 4 x + 2 y + \frac{5}{2}) (\frac{5}{2}) = {(- 2 x + y + \frac{5}{2})}^{2} \\ \Rightarrow 3 x^{2} - 8 x y - 3 y^{2} = 0 \Rightarrow 3 x + y = 0, x - 3 y = 0 \end{aligned}$

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Definition of a circle

Remember concepts with our Masterclasses.

If 3x+y=0 is a tangent to the circle with centre at the point (2, -1), then the equation of the other tangent to the circle from the origin, is

Ready to Test Your Skills?

free study material

If $3 x + y = 0$ is a tangent to the circle with centre at the point (2, -1), then the equation of the other tangent to the circle from the origin, is