The only condition for a line y = mx + c to be tangent on hyperbola x2 + y2 is;
c = ± √a2m2 – b2
Example: Find the point of contact of a line 5x + 2y = 9 touching the hyperbola x2 – 9y2 = 9.
Solution: The given line is 5x + 12y = 9 ….(i)
The given hyperbola is x2 – 9y2 = 9. ….(ii)
9x2 – 90x +225 = 0
X2 – 10 x + 25 = 0
(x – 5)2 = 0
X = 5, 5 are the two coincident roots.
Putting x = 5 in (i) we get y = -(4/3)
The point of contact is (5, -4/3)
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