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10-1.Circle and System of Circles
medium
The equations of tangents to the circle ${x^2} + {y^2} - 22x - 4y + 25 = 0$ which are perpendicular to the line $5x + 12y + 8 = 0$ are
A
$12x - 5y + 8 = 0$, $12x - 5y = 252$
B
$12x - 5y = 0,\,\,12x - 5y = 252$
C
$12x - 5y - 8 = 0,\,12x - 5y + 252 = 0$
D
None of these
Solution
(a) Equation of line perpendicular to $5x + 12y + 8 = 0$ is $12x – 5y + k = 0$.
Now it is a tangent to the circle, if
Radius of circle = Distance of line from centre of circle
$\sqrt {121 + 4 – 25} = \left| {\frac{{12(11) – 5(2) + k}}{{\sqrt {144 + 25} }}} \right|$
$\Rightarrow k = 8$ or $-252.$
Hence equations of tangents are $12x – 5y + 8 = 0$ and $12x – 5y = 252$.
Standard 11
Mathematics
