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જો રેખા $(x + g) cos\ \theta + (y +f) sin\theta = k$ વર્તૂળ $x^2 + y^2 + 2gx + 2fy + c =0$ , ને સ્પર્શેં, તો
$g^2 + f^2 = k^2 + c^2$
$g^2 + f^2 = k + c$
$g^2 + f^2 = k^2 + c$
એકપણ નહિ
Solution
Given circle
$x ^2+ y ^2+2 gx +2 gy + c =0$
where centre of circle $C (- g ,- f )$
and radius $r =\sqrt{g^2+ f ^2- c }$
Given line $x \cos \theta+y \sin \theta+(g \cos \theta+f \sin \theta-k=0)$
If line is tangent to circle then
$p=\left|\frac{a x_1+b y_1+c}{\sqrt{a^2+b^2}}\right|=r$
Hence perpendicular ditance of given line from centre $C (- g ,- f )$
$r=\left|\frac{g \cos \theta-f \sin \theta+g \cos \theta+f \sin \theta-k}{\sqrt{\sin ^2 \theta+\cos ^2 \theta}}\right|$
$\sqrt{g^2+f^2-c}=\left|\frac{-k}{\sqrt{1}}\right|$ here $\left[\sin ^2 \theta+\cos ^2 \theta=1\right]$
$\sqrt{g^2+f^2-c}=k$
On squaring both sides
$g^2+f^2-c=k^2$
$g^2+f^2=c+k^2$
