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10-1.Circle and System of Circles
easy
બિંદુ $(2, 3)$ માંથી વર્તૂળ $2\ (x^2 + y^2) - 7x + 9y - 11 = 0$ પર દોરેલા સ્પર્શકની લંબાઈ :
A
$18$
B
$14$
C
$\sqrt {14} $
D
$\sqrt {28} $
Solution
REF.Image
$x^2+y^2-\frac{7 x}{2}+\frac{9 y}{2}-\frac{11}{2}=0$
$A \equiv\left(\frac{7}{4}, \frac{-9}{4}\right)$
$\text { Radius }=\sqrt{\frac{49+81}{16}+\frac{11}{2}}=\sqrt{\frac{65}{8}+\frac{44}{8}}$
$=\sqrt{\frac{109}{8}}$
$A C=\sqrt{\left(\frac{2-7}{4}\right)^2+\left(3+\frac{9}{4}\right)^2}$
$\sqrt{\frac{1}{16}+\frac{441}{16}}=\sqrt{\frac{221}{8}}$
$B C=\sqrt{(A C)^2-(A B)^2}$
$=\sqrt{\frac{221-109}{8}}=\sqrt{\frac{112}{8}}$
$=\sqrt{14} \text { units }$
Standard 11
Mathematics
