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A variable chord x*cos(a) + y*sin(a) = P of the hyperbola x²/a² - y²/b² = 1 (with b > a) subtends a right angle at the centre. Show that this chord always remains tangent to a fixed circle and find the radius of that circle.

1. ab / sqrt(a² + b²)
2. ab / sqrt(b² - a²)
3. ab / sqrt(a² - b²)
4. None of these

Correct answer: ab / sqrt(b² - a²)

Solution

Homogenising and applying the right-angle condition gives a fixed value for P, which is the perpendicular distance from the origin to the chord, establishing that the chord is always tangent to a circle of radius ab/sqrt(b²-a²).

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