First slide

Definition of a circle

Question

The equation of any tangent to the circle $x^{2} + y^{2} - 2 x + 4 y - 4 = 0$ is

Moderate

A

$y = m (x - 1)^{2} + 3 \sqrt{1 + m^{2}} - 2$
B

$y = m x + 3 \sqrt{1 + m^{2}}$
C

$y = m x + 3 \sqrt{1 + m^{2}} - 2$
D

none of these

Solution

We have,
$\begin{array}{l} x^{2} + y^{2} - 2 x + 4 y - 4 = 0 \\ \Rightarrow (x^{2} - 2 x + 1) + (y^{2} + 4 y + 4) = 3^{2} \\ \Rightarrow (x - 1)^{2} + (y + 2)^{2} = 3^{2} \end{array}$
The equation of any tangent of slope mis given by
$y + 2 = m (x - 1) \pm 3 \sqrt{1 + m^{2}}$
Thus, option (a) is correct.

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