Question

The circles $r=a\cos \theta +b\sin \theta$ and $r=c\cos \theta +d\sin \theta$ touch each other then

• A. $ab=cd$
• B. $ab+cd=0$
• C. $ad=bc$
• D. $ad+bc=0$

Answer 1

Answer: (C)

$ad=bc$

Put $\cos \theta =\frac{x}{r}$ & $\sin \theta =\frac{y}{r}$ in $r=a\cos \theta +b\sin \theta$
where, $r=\sqrt{x^2+y^2}$
$r=\frac{ax}{r}+\frac{by}{r}$
$\Rightarrow r^2=ax+by$
$C_1:\Rightarrow x^2+y^2-ax-by=0$
Similarly for $r=c\cos \theta +d\sin \theta$
We get $C_2:x^2+y^2-cx-dy=0$
Condition if the circles touch each other is $\frac{g}{g^{\prime }}=\frac{f}{f^{\prime }}$
Therefore, $\frac{a}{c}=\frac{b}{d}$
$\Rightarrow ad=bc$
Ans: C