Question

The number of points from where a pair of perpendicular tangents can be drawn to the hyperbola, x²sec²α – y²cosec²α = 1, `α∈(0, π/4)` are ______.

Options

• 0

• 1

• 2

• infinite

Answer 1

The number of points from where a pair of perpendicular tangents can be drawn to the hyperbola, x²sec²α – y²cosec²α = 1, `α∈(0, π/4)` are infinite.

Explanation:

Given, x²sec²α – y²cosec²α = 1

⇒ `x^2/(cos^2α) - y^2/(sin^2α)` = 1

Here, α² = cos²α, b² = sin²α

Then, equation of its director circle is x² + y² = a² – b²

⇒ x² + y² = cos²α – sin²α

= 2cos²α – 1

Director circle will exist as

2cos²α > 1 ...`("when", α ∈(0, π/4))`

2cos²α – 1 > 0

So, there will exists infinite points.