First slide

Definition of a circle

Question

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

Moderate

A

$x - 3 y = 0$
B

$x + 3 y = 0$
C

$3 x - y = 0$
D

$2 x + y = 0$

Solution

Let y = mx be a tangent drawn from the origin to the circle having its centre at (2, -1) and touching $3 x + y = 0$
Then,
$\begin{array}{l} |\frac{2 m + 1}{\sqrt{m^{2} + 1}}| = |\frac{6 - 1}{\sqrt{9 + 1}}| \\ \Rightarrow 2 (2 m + 1)^{2} = 5 (m^{2} + 1) \\ \Rightarrow 3 m^{2} + 8 m - 3 = 0 \Rightarrow (3 m - 1) (m + 3) = 0 \Rightarrow m = - 3, \frac{1}{3} \end{array}$
Thus, the equation of the tangents drawn from the origin are
$y = - 3 x and y = x / 3$

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