Question

The value of p so that the straight line xcosα + ysinα – p = 0 may touch the circle x² + y² – 2ax cosα – 2by sinα – a²sin²α = 0 is ______.

Options

• `acos^2α + bsin^2α - sqrt(a^2 + b^2sin^2α)`

• `acos^2α - bsin^2α - sqrt(a^2 + b^2sin^2α)`

• `acos^2α + bsin^2α - sqrt(a^2 - b^2sin^2α)`

• `acos^2α - bsin^2α + sqrt(a^2 - b^2sin^2α)`

Answer 1

The value of p so that the straight line xcosα + ysinα – p = 0 may touch the circle x² + y² – 2ax cosα – 2by sinα – a²sin²α = 0 is `underlinebb(acos^2α + bsin^2α - sqrt(a^2 + b^2sin^2α))`.

Explanation:

xcosα + ysinα – P = 0 touches the circle

x² + y² – 2ax cosα – 2by sinα – a²sin²α = 0

∴ Centre of circle = (acosα, bsinα)

Radius of circle = `sqrt(a^2cos^2α + b^2sin^2α + a^2sin^2α)`

= `sqrt(a^2 + b^2sin^2α)`

Perpendicular drawn from centre to line is equal to the radius

  i.e. `(acos^2α + bsin^2α - p)/(sqrt(sin^2α + cos^2α)) = sqrt(a^2 + b^2sin^2α)`

⇒ acos²α + bsin²α – p = `sqrt(a^2 + b^2sin^2α)`

⇒ p = `acos^2α + bsin^2α - sqrt(a^2 + b^2sin^2α)`