Choose the correct option regarding the following statements : Statement I The length of common chord of the…
Choose the correct option regarding the following statements :
Statement I The length of common chord of the circles $x^2+y^2+a x+b y+c=0$ and
$x^2+y^2+b x+a y+c=0$ equals $\frac{\sqrt{(a+b)^2-8 c}}{2}$
Statement II If two circles intersect at two distinct points, then their radical axis is their common chord.
Both statements are true and statement-II is a correct explanation for statement-I.
Both statements are true but statement-II is not a correct explanation for statement-I.
Statement-I is true, Statement-II is false.
Statement-I is false, Statement-II is true.
Solution
Statement I The length of common chord of the circles $x^2+y^2+a x+b y+c=0$ and
$x^2+y^2+b x+a y+c=0$ equals $\frac{\sqrt{(a+b)^2-8 c}}{2}$
Let the circles $\mathcal{c}_1$ be the center of
$x^2+y^2+a x+b y+c=0$ ...(i)
and $c_2$ be the center of
$x^2+y^2+b x+a y+c=0$ ...(ii)
$\begin{aligned} & \therefore \quad c_1=\left(\frac{-a}{2}, \frac{-b}{2}\right) \\ & \text { and } c_2=\left(\frac{-b}{2}, \frac{-a}{2}\right)\end{aligned}$
$r_1=\sqrt{\frac{a^2}{4}+\frac{b^2}{4}-c}$
$\Rightarrow \quad r_1=\frac{1}{2} \sqrt{a^2+b^2-4 c}$
$\begin{aligned} & r_2=\sqrt{\frac{b^2}{4}+\frac{a^2}{4}-c} \\ & r_2=\frac{1}{2} \sqrt{a^2+b^2-4 c}\end{aligned}$
$\because$ Radius of both circles is same.
$c_1 c_2=\sqrt{\left(\frac{-a}{2}+\frac{b}{2}\right)^2+\left(\frac{-b}{2}+\frac{a}{2}\right)^2}$
$\begin{aligned} & \sqrt{\frac{(a-b)^2}{4}+\frac{(a-b)^2}{4}} \\ & c_1 c_2=\frac{a-b}{\sqrt{2}}\end{aligned}$
Let $A B$ be common chord and $M$ is its mid-point.
$\therefore \quad C_1 M=\frac{c_1 c_2}{2}=\frac{a-b}{2 \sqrt{2}}$
$\because A c_1 M$ forms a right angle triangle.
$\therefore \quad A M=\sqrt{r_1^2-\left(c_1 M\right)^2}$
$\begin{aligned} & =\sqrt{\frac{a^2+b^2-4 c}{4}-\frac{a^2+b^2-2 a b}{8}} \\ & =\frac{\sqrt{(a+b)^2-8 c}}{2 \sqrt{2}}\end{aligned}$
$\therefore \quad A B=2 A M=\frac{\sqrt{(a+b)^2-8 c}}{\sqrt{2}}$
$\Rightarrow$ Statement -1 is false.
We know that if two circles intersect at two distinct points, then their radical axis is their common chord.
$\Rightarrow$ Statement- 2 is true
