Angle between two tangents from origin to circle {(x - 7)^2} + {(y + 1)^2} = 25 is

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10-1.Circle and System of Circles

hard

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

A

$0$

B

$\frac{\pi }{3}$

C

$\frac{\pi }{6}$

D

$\frac{\pi }{2}$

Solution

(d) Any line through $(0, 0)$ be $y – mx = 0$ and it is a tangent to circle ${(x – 7)^2} + {(y + 1)^2} = 25$, if

$\frac{{ – 1 – 7m}}{{\sqrt {1 + {m^2}} }} = 5 \Rightarrow m = \frac{3}{4},\; – \frac{4}{3}$.

Therefore, the product of both the slopes is $-1.$

i.e., $\frac{3}{4} \times – \frac{4}{3} = – 1$.

Hence the angle between the two tangents is $\frac{\pi }{2}$.

Standard 11

Mathematics

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