Question

The abscissae of two points A and B are the roots of the equation x² + 2ax – b² = 0, and their ordinates are the roots of the equation x² + 2px – q² = 0. The radius of the circle with AB as diameter is ______.

Options

• `sqrt(a^2 + b^2 + p^2 + q^2)`

• `sqrt(a^2 + p^2)`

• `sqrt(b^2 + q^2)`

• `sqrt(a^2 + b^2 - p^2 - q^2)`

Answer 1

The abscissae of two points A and B are the roots of the equation x² + 2ax – b² = 0, and their ordinates are the roots of the equation x² + 2px – q² = 0. The radius of the circle with AB as diameter is `underlinebb(sqrt(a^2 + b^2 + p^2 + q^2))`.

Explanation:

Let A ≡ (x₁, y₁) and B ≡ (x₂, y₂) since x₁, x₂ are roots of x² + 2ax – b² = 0 therefore

x₁ + x₂ = –2a

x₁x₂ = –b²

Also y₁, y₂ are roots of x² + 2px – q² = 0

∴ y₁ + y₂ = –2p

y₁y₂ = –q²

∴ AB = `sqrt((x_1 - x_2)^2 + (y_1 - y_2)^2`

= `sqrt({(x_1 + x_2)^2 - 4x_1x_2} + {(y_1 + y_2)^2 - 4y_1y_2}`

= `sqrt(4a^2 + 4b^2 + 4p^2 + 4q^2)`

= `2sqrt(a^2 + b^2 + p^2 + q^2)`

∴ Radius = `"Diameter"/2`

= `(2sqrt(a^2 + b^2 + p^2 + q^2))/2`

= `sqrt(a^2 + b^2 + p^2 + q^2)`