Question

If the lines a₁x +b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 cut the co-ordinate axes in concyclic points, then-

• A. a₁b₁ = a₂b₂
• B. $\frac { a _ { 1 } } { a _ { 2 } } = \frac { b _ { 1 } } { b _ { 2 } }$
• C. a₁ + a₂ = b₁ + b₂
• D. a₁a₂ = b₁b₂

Answer 1

Answer: (D)

a₁a₂ = b₁b₂

Answer 2

Let the given lines be L₁ ŗ a₁x + b₁y + c₁ = 0 and L₂ ŗ a₂x + b₂y + c₂ = 0, suppose L₁ meets the co-ordinates axes at A and B and L₂ meets at C & D. Then co-ordinates of A,B,C,D are

A $\left( - \frac { c _ { 1 } } { a _ { 1 } } , 0 \right)$ , B $\left( 0 , - \frac { \mathrm { c } _ { 1 } } { \mathrm {~b} _ { 1 } } \right)$ , C

and D

Since A, B, C, D are concyclic, therefore

OA . OC = OD . OB

Ž $\left( - \frac { c _ { 1 } } { a _ { 1 } } \right)$ $\left( - \frac { c _ { 2 } } { a _ { 2 } } \right)$ = $\left( - \frac { c _ { 2 } } { b _ { 2 } } \right)$ $\left( - \frac { c _ { 1 } } { b _ { 1 } } \right)$

Ž a₁a₂ = b₁b₂ .