CircleHard
Question
A variable circle is drawn to touch the x-axis at the origin. The locus of the pole of the straight line lx + my + n = 0 w.r.t. the variable circle has the equation
Options
A.x(my - n) - ly2 = 0
B.x(my + n) - ly2 = 0
C.x(my - n) + ly2 = 0
D.none
Solution
Equation of variable circle which touch the x-axis at origin is
x2 + y2 + λy = 0
Let the pole of the above circle be P(h, k)
Equation of polar is
hx + ky +
(y + k) = 0
hx + (k +
) y + 
= 0 .....(1)
and the equation of given polar is
x + my + n = 0 .....(2)
comparing (1) and (2)
⇒ mh = lk +
and nh = 
⇒ mh = + lk +
⇒ mhk = lk2 + nh
∴ x(my - n) - ly2 = 0
x2 + y2 + λy = 0
Let the pole of the above circle be P(h, k)
Equation of polar is
hx + ky +
(y + k) = 0hx + (k +
) y + = 0 .....(1)and the equation of given polar is
x + my + n = 0 .....(2)
comparing (1) and (2)
⇒ mh = lk +
and nh = ⇒ mh = + lk +
⇒ mhk = lk2 + nh∴ x(my - n) - ly2 = 0
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