Question

If the origin be one limiting point of a system of co-axial circles of which x² +y² +3x+4y+25=0 is a member, find the other limiting point.

Answer 1

Equation of circle with origin (0,0) as limiting point is x² +y² =0

Given that one member of system of co-axial circle is x² +y²+3x+4y+25 = 0

∴ The system of co-axial circles is

x² +y² + \(\frac{3}{1+λ }x+\frac{4}{1+λ}y+\frac{25}{1+λ }=0\)

centre \(\left(\frac{-3}{2(1+λ)},\frac{-2}{1+λ }\right)\)

radius \(\frac{9}{4(1+λ)^2}+\frac{4}{(1+λ)^2}-\frac{25}{1+λ}=0\)

\(\frac{25}{4(1+λ)^2}-\frac{25}{1+λ}=0\)

1 – 4(1+λ) = 0

1 + λ = 1/4

λ = 1/4 – 1 = –3/4

∴centre (–6,–8) is the other limiting point of the system .