HyperbolaHard
Question
If x = 9 is the chord of contact of the hyperbola x2 - y2 = 9, then the equation of the corresponding pair of tangents is
Options
A.9x2 - 8y2 + 18 x - 9 = 0
B.9x2 - 8y2 - 1 8 x + 9 = 0
C.9x2 - 8y2 -18x - 9 = 0
D.9x2 - 8y2 + 18x + 9 = 0
Solution
Let (h, k) be point whose chontact with respect to hyperbola x2 - y2 = 9 is x = 9.
We know that hord of contact of (h, k ) with respect to hyerbola x2 - y2 = 9 is T = 0
⇒ h.x + k(- y) - 9 = 0
∴ hx + ky - 9 = 0
But it is the equation of the line x = 9
This is possible when h = 1, k = 0 (by comparing both equations).
Again equation of pair of tangents is T2 = SS1
⇒ (x - 9)2 = (x2 - y2 - 9)(12 - 02 - 9)
⇒ x2 -18x +81= (x2 - y2 - 9)(-8)
⇒ x2 - 18x + 81 = - 8x2 + 8y2 + 72
⇒ 9x2 - 8y2 - 18x + 9 = 0
We know that hord of contact of (h, k ) with respect to hyerbola x2 - y2 = 9 is T = 0
⇒ h.x + k(- y) - 9 = 0
∴ hx + ky - 9 = 0
But it is the equation of the line x = 9
This is possible when h = 1, k = 0 (by comparing both equations).
Again equation of pair of tangents is T2 = SS1
⇒ (x - 9)2 = (x2 - y2 - 9)(12 - 02 - 9)
⇒ x2 -18x +81= (x2 - y2 - 9)(-8)
⇒ x2 - 18x + 81 = - 8x2 + 8y2 + 72
⇒ 9x2 - 8y2 - 18x + 9 = 0
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