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10-1.Circle and System of Circles
normal
Pair of tangents are drawn from every point on the line $3x + 4y = 12$ on the circle $x^2 + y^2 = 4$. Their variable chord of contact always passes through a fixed point whose co-ordinates are
A
$\left( {\frac{4}{3}\,,\,\frac{3}{4}} \right)$
B
$\left( {\frac{3}{4}\,,\,\frac{3}{4}} \right)$
C
$(1, 1)$
D
$\left( {1\,,\,\frac{4}{3}} \right)$
Solution
Let $P\left(x_{1}, y_{1}\right)$ be a point on the line $3 x+4 y=12$
Equation of variable chord of contact of $P\left(x_{1}, y_{1}\right)$ w.r.t circle $x^{2}+y^{2}=4$ (figure)
$x x_{1}+y y_{1}-4=0$
Also $3 x_{1}+4 y_{1}-12=0 \Rightarrow x_{1}+\frac{4}{3} y_{1}-4=0$
Comparing (1) and (2) , we get $x=1 ; y=\frac{4}{3}$
$\therefore$ Variable chord of contact always passes through $\left(1, \frac{4}{3}\right)$
Standard 11
Mathematics
