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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
Similar Questions
Match the statements in Column $I$ with the properties Column $II$ and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the $ORS$.
| Column $I$ | Column $II$ |
| $(A)$ Two intersecting circles | $(p)$ have a common tangent |
| $(B)$ Two mutually external circles | $(q)$ have a common normal |
| $(C)$ two circles, one strictly inside the other | $(r)$ do not have a common tangent |
| $(D)$ two branches of a hyperbola | $(s)$ do not have a common normal |
