Paragraph: A tangent PT is drawn to the circle x 2 + y 2 = 4 at the point P ( 3 , 1 ) . A straight line L , perpendicular to PT is a tangent to the circle ( x − 3 ) 2 + y 2 − 1 . Question: A common

Question

Paragraph: A tangent PT is drawn to the circle x 2 + y 2 = 4 at the point P ( 3 ​ , 1 ) . A straight line L , perpendicular to PT is a tangent to the circle ( x − 3 ) 2 + y 2 − 1 . Question: A common tangent of the two circles is

TestHub · Accepted Answer

From the figure it is clear that the intersection point of two direct common tangents lies on $x$ -axis. Also $Δ P T_{1} C_{1} \sim Δ P T_{2} C_{2}$ $∴ P C_{1} : P C_{2} = 2 : 1$ or $P$ divides $C_{1} C_{2}$ in the ratio $2 : 1$ externally $∴$ Coordinates of $P$ are $(6, 0)$ . Let the equation of tangent through $P$ be $y = m (x - 6)$ Asit touches $x^{2} + y^{2} = 4$ $∴ \frac{6 m}{m ^{2} + 1} = 2 \Rightarrow 36 m^{2} = 4 (m^{2} + 1)$ $∴ m = \pm \frac{1}{2 2}$ $∴$ Equations of common tangents are $y = \pm \frac{1}{2 2} (x - 6)$ Also $x = 2$ is the common tangent to the two circles.

Mathematics - Circle Question with Solution | TestHub

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