JEE Advanced | 2015CircleHard
Question
Consider the hyperbola H : x2 − y2 = 1 and a circle S with center N(x2, 0). Suppose that H and S touch each other at a point P(x1, y1) with x1 > 1 and y1 > 0. The common tangent to H and S at P intersects the x-axis at point M. If (l, m) is the centroid of the triangle ᐃPMN, then the correct expression (s) is(are)
Options
A.
for x1 > 1
for x1 > 1B.
for x1 > 1
for x1 > 1C.
for x1 > 1
for x1 > 1D.
for y1 > 1
for y1 > 1Solution
Given H : x2 - y2 = 1
Now, equation of family of circle touching
hyperbola at (x1, y1) is
(x - x1)2 + (y - y1)2 + λ(xx1 - yy1 - 1) = 0
Now, it centre is (x2, 0)
∴ 2y1 + λy1 = 0 ⇒ λ = - 2
∴ x2 =
∴ P ≡ (x1,
)
N ≡ (2x1, 0)
& M ≡
∴
x1 > 1
Also m =
y1 > 0
∴ (A), (B) & (D)
Now, equation of family of circle touching
hyperbola at (x1, y1) is
(x - x1)2 + (y - y1)2 + λ(xx1 - yy1 - 1) = 0
Now, it centre is (x2, 0)
∴ 2y1 + λy1 = 0 ⇒ λ = - 2
∴ x2 =
∴ P ≡ (x1,
)N ≡ (2x1, 0)
& M ≡
∴
x1 > 1Also m =
y1 > 0∴ (A), (B) & (D)
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