Suppose $A(2,3)$ and $B$ are the points of intersections of two circles. The points $P$ lying on one circle and $Q$ lying on the other circle are such that $B P$ and $B Q$ constitute the diameters of the circles. If the slopes of the radical axis and $P Q$ are $3 / 4$ and $a / b$ respectively,…

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Suppose $A(2,3)$ and $B$ are the points of intersections of two circles. The points $P$ lying on one circle and $Q$ lying on the other circle are such that $B P$ and $B Q$ constitute the diameters of the circles. If the slopes of the radical axis and $P Q$ are $3 / 4$ and $a / b$ respectively, then the value of $3 a+4 b$ is

Aicharya · Accepted Answer

$C_1$ and $C_2$ are centre of circles.

$ \therefore \quad B C_1=C_1 P, B C_2=C_2 Q . $ Consider $\triangle B P Q$ $C_1 C_2$ divides sides $B P$ and $B Q$ in same ratio. $ \begin{array}{ll} \therefore & P Q \| C_1 C_2 \\ \Rightarrow & P Q \perp A B \end{array} $ As line $A B$ is also radical axis, product of slopes of $P Q$ and radical axis -1 . $ \begin{array}{ll} \therefore & \frac{3}{4} \times \frac{a}{b}=-1 \\ \Rightarrow & 3 a+4 b=0 \end{array} $

Suppose $A(2,3)$ and $B$ are the points of intersections of two circles. The points $P$ lying on one circle…

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