Question

The angle between the tangents to the circle x² + y² = 25 from the point (7, -1) is ______.

पर्याय

• `pi/3`

• `pi/6`

• `pi/4`

• `pi/2`

Answer 1

The angle between the tangents to the circle x² + y² = 25 from the point (7, -1) is `underline(pi/2)`.

Explanation:

The equation of the circle is x² + y² = 25

∴ radius a = 5

Let P = (7, -1)

Let the equation of a tangent to the given circle from P be

y = mx ± a`sqrt(1 + "m"^2)`

∴ y = mx ± 5`sqrt(1 + "m"^2)`

As P(7, - 1) lies on tangent, we have

-1 = 7m ± 5`sqrt(1 + "m"^2)`

⇒ (7m + 1)² = 25(1 + m²)

⇒ 49m² + 1 + 14m = 25 + 25m²

⇒ 24m² + 14m - 24 = 0

⇒ 12m² + 7m - 12 = 0

⇒ m = `3/4, - 4/3`

Let m₁ = `3/4` and m₂ = `-4/3`

Then, m₁m₂ = - 1

Hence, the angle between the tangents is `pi/2`