Question
Question: If 4 – 5m<sup>2</sup> + 6l + 1 = 0, then the line lx + my + 1 = 0 touches a fixed circle. Then the e...
If 4 – 5m2 + 6l + 1 = 0, then the line lx + my + 1 = 0 touches a fixed circle. Then the equation of circle is –
A
(x + 3)2 + (y – 0)2 = 5
B
(x – 3)2 + (y – 0)2 = 5
C
(x – 3)2 + (y – 0)2 = 0
D
None of these
Answer
(x – 3)2 + (y – 0)2 = 5
Explanation
Solution
Let the line lx + my + 1 = 0 touch the circle
(x – a)2 + (y – b)2 = r2
Then 
= r
Ž (la + mb + 1)2 = r2 (l2 + m2)
l2 (a2 – r2) + m2(b2 – r2) + 2ablm + 2al + 2bm + 1 = 0
On comparing this with the equation 4l2 – 5m2 + 6l + 1 = 0,
we have
a2 – r2 = 4, b2 – r2 = –5, 2ab = 0, 2a = 6, 2b = 0
Ž a = 3, b = 0 and r2 = 5
Clearly, above equations are consistent. Thus, the line touches the fixed circle (x – 3)2 + (y – 0)2 = 5.