If the circles x^2+y^2+2ax+cy+a=0 and x^2+y^2-3ax+dy-1=0 intersect in two distinct points …

Question

If the circles $x^2+y^2+2ax+cy+a=0$ and $x^2+y^2-3ax+dy-1=0$ intersect in two distinct points $P$ and $Q$, then the line $5x+by-a=0$ passes through $P$ and $Q$ for:

Accepted Answer

${s_1} = {x^2} + {y^2} + 2ax + cy + a = 0$

${s_2} = {x^2} + {y^2} - 3ax + dy - 1 = 0$

Equation of common chord of circles ${s_1}$ and ${s_2}$ is

given by ${s_1} - {s_2} = 0$

$$ \Rightarrow 5ax + \left( {c - d} \right)y + a + 1 = 0$$

Given that $5x + by - a = 0$ passes through $P$ and $Q$

$$\therefore$$ The two equations should represent the same line

$$ \Rightarrow {a \over 1} = {{c - d} \over b} = {{a + 1} \over { - a}}$$

$$ \Rightarrow a + 1 =  - {a^2}$$

${a^2} + a + 1 = 0$

No real value of $a.$

Solutions · 1