Question

If the line a1x+b1y+c1=0 and a2x+b2y+c2=0 cut the coordinate axis at concyclic points ,then prove that a1a2=b1b2.

Answer 1

Points are said to be Concyclic if they lie on the same circle or their distance from the center are same.

The two lines are$a_{1} x+b_{1} y+c_{1}=0$ =0 and $a_{2} x+b_{2} y+c_{2}=0$=0

It cuts the coordinate axes at $A(-\frac{c_{1}}{a_{1}},0),B(0,-\frac{c_{1}}{b_{1}}),C(-\frac{c_{2}}{a_{2}},0),D(0,-\frac{c_{2}}{b_{2}})$

As distance of these points from origin are same.

OA=$\frac{{c_{1}}^2}{{a_{1}}^2}$.....(1)

OB=$\frac{{c_{1}}^2}{{b_{1}}^2}$......(2)

OC=$\frac{{c_{2}}^2}{{a_{2}}^2}$......(3)

OD=$\frac{{c_{2}}^2}{{b_{2}}^2}$...........(4)

Equating 1 and 2, we get

$a_{1}=b_{1}$........(5)

Equating 3 and 4, we get

$a_{2}=b_{2}$.......(6)

Multiplying (5) and (6) i.e LHS by LHS and RHS by RHS, we get

$a_{1}a_{2}=b_{1}b_{2}$

Hence proved