7.6.5 Tangent and Normal at a point on Hyperbola:

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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