First slide

Definition of a circle

Question

The points of contact of tangents to the circle $x^{2} + y^{2} = 25$ which are inclined at an angle of $30 °$ to the $x -$ axis are

Moderate

A

$(\pm 5 / 2, \pm 1 / 2)$
B

$(\pm 1 / 2, \pm 5 / 2)$
C

$(\mp 5 / 2, \mp 1 / 2)$
D

none of these

Solution

The equation of a tangent of slope $m$ to the circle $x^{2} + y^{2} = a^{2}$ is $y = m x \pm a \sqrt{1 + m^{2}}$ and the coordinates of point of contact are
$(\mp \frac{a m}{\sqrt{1 + m^{2}}}, \pm \frac{a}{\sqrt{1 + m^{2}}})$
Here, $a = 5$ and $m = \tan 30 = 1 / \sqrt{3} .$
So, the coordinates of the points of contact are $(\mp \frac{5}{2}, \pm \frac{5 \sqrt{3}}{2})$

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