The equation of a tangent at a point P(x1, y1) of the hyperbola x2 -y2 = 1 is given by ;
a2 b2
xx1 - yy1 = 1
a2 b2
and the equation of normal at this point is given by;
x – x1 - y - y1 = 1
x1 /a2 y1/ b2
Example: Find the equation of tangent and normal to the hyperbola 16x2 – 25y2 = 400 at the point (3, 2).
Solution: Equation of the given hyperbola
x2- y2 = 1
25 16
Therefore, the equation of tangent to the hyperbola at (3,2) is;
xx1 - yy1 = 1
a2 b2
a2 = 9 and b2 = 4; x1 = 3 and y1 = 2
3x - 2y = 1 or x - y = 1
9 4 3 2
Or
2x – 3y = 6
The equation of the normal at (3, 2) is
(x – 3) + (y – 2)
3/9 2/4
3x + 2y = 5
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