Question

the product of length of perpendiculars drawn from foci on any tangent to the hyperbola x^2/a^2 - y^2/b^2=1 is

Answer 1

Given:

The equation of the hyperbola is x^2/a^2 - y^2/b^2=1

To find:

The product of the length of perpendiculars drawn from foci on any tangent to the hyperbola x^2/a^2 - y^2/b^2=1 is

Solution:

From given, we have,

The equation of the hyperbola is x^2/a^2 - y^2/b^2 = 1

The hyperbola is on the minor axis is  y = b

The two foci are given by,

S1(ae,0) and S2(-ae,0)

Let the equation of tangent be ax+by+c = 0

Perpendicular length from foci to tangent is given by,  

S1 = |b/√(-1)| = b

and S2 = |b/√(-1)| = b

The product of foci is = S1 × S2 = b²