Q.11. To find radius of a circle under given conditions

Question 11:
f(x,y)=0 is a circle such that f(0,λ)=0 and f(λ,0)=0 have equal roots and f(1,1)=-2 then radius of the circle is________?


Answer 11:
x^2 + y^2 + 2gx + 2fy + c = 0

f(0,λ)=0 and f(λ,0)=0 have equal roots
=> 0 + λ^2 + 2g*(0) + 2f*(λ) + c = 0 and λ^2 + 0 + 2g*(λ) + 2f*0 + c = 0
i.e., λ^2 + 2gλ + c = 0 and λ^2 + 2fλ + c = 0 have equal roots
=> discriminants of both the quadratics = 0
=> g^2 - c = 0 and f^2 - c = 0
=> g^2 = f^2 = c
=> equation of the circle is
x^2 + y^2 ± 2x√c ± 2y√c + c = 0
f(1,1) = -2
=> 1 + 1 ± 2√c ± 2√c + c = -2
=> c + 4 = ± 4√c
=> c^2 + 8c + 16 = 16c
=> c^2 - 8c + 16 = 0
=> (c - 4)^2 = 0 => c = 4
=> radius of the circle = √(g^2 + f^2 - c)
= √(c + c - c) = √c
= √4
= 2.

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