CircleHard
Question
If the line 3x - 4y - k = 0 (k > 0) touches the circle x2 + y2 - 4x - 8y - 5 = 0 at (a, b) then k + a + b is equal to :-
Options
A.20
B.22
C.-30
D.-28
Solution
Since, the given line touches the given circle, the length of the perpendicular from the centre (2, 4) of the circle to the line 3x - 4y - k = 0 is equal to the radius 
= 5 of the circle.
∴
= ± 5
⇒ k = 15 [∵ k > 0]
hence equation of tangent is 3x - 4y - 15 = 0 ..... (1)
Let equation of normal to circle 4x + 3y = λ
It passes through centre (2, 4)
⇒ λ = 20
hence equation of normal is
4x + 3y = 20 .... (2)
Solve (1) & (2)
a = 5, b = 0
k + a + b = 15 + 5 + 0 = 20
= 5 of the circle.∴
= ± 5⇒ k = 15 [∵ k > 0]
hence equation of tangent is 3x - 4y - 15 = 0 ..... (1)
Let equation of normal to circle 4x + 3y = λ
It passes through centre (2, 4)
⇒ λ = 20
hence equation of normal is
4x + 3y = 20 .... (2)
Solve (1) & (2)
a = 5, b = 0
k + a + b = 15 + 5 + 0 = 20
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