First slide

Definition of a circle

Question

The angle between two tangents drawn from
the origin to the circle $(x - 7)^{2} + (y + 1)^{2} = 25,$ is

Moderate

A

$\frac{π}{4}$
B

$\frac{π}{3}$
C

$\frac{π}{2}$
D

$\frac{2 π}{3}$

Solution

The equation of any line through the origin ( 0, O) is
$y = m x$
If it is a tangent to the circle $(x - 7)^{2} + (y + 1)^{2} = 5^{2}$ then,
$\begin{array}{l} |\frac{7 m + 1}{\sqrt{m^{2} + 1}}| = 5 \\ \Rightarrow (7 m + 1)^{2} = 25 (m^{2} + 1) \Rightarrow 24 m^{2} + 14 m - 24 = 0 \end{array}$
This equation, being a quadratic inm, gives two values of
$m,$ say $m^{1}$ and $m^{2}$ . These two values of mare the slopes
of the tangents drawn from he origin to the given circle.
From (i), we have $m_{1} m_{2} = - 1$
Hence, the two tangents are perpendicular
Hence, the two tangents are perpendicular
$(x - 7)^{2} + (y + 1)^{2} = 50$ of the given circle.
Hence, required angle is $π / 2$

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