Question

The abscissae of the two points A and B are the roots of the equation $x^2 + 2ax - b^2 = 0$ and their ordinates are roots of the equation $y^2 + 2py - q^2 = 0$. Then the equation of the circle with AB as diameter is given by

• A. $x^2 + y^2 - 2ax - 2py + (b^2 + q^2) = 0$
• B. $x^2 + y^2 - 2ax - 2py - (b^2 + q^2) = 0$
• C. $x^2 + y^2 + 2ax + 2py + (b^2 + q^2) = 0$
• D. $x^2 + y^2 + 2ax + 2py - (b^2 + q^2) = 0$

Answer 1

Answer: (D)

Option D

Answer 2

Let the coordinates of points A and B be A(x₁, y₁) and B(x₂, y₂).

• The quadratic in x:

  $x^2 + 2ax - b^2 = (x - x₁)(x - x₂)$
  $\Rightarrow x₁ + x₂ = -2a$, and $x₁x₂ = -b²$.

• The quadratic in y:

  $y^2 + 2py - q^2 = (y - y₁)(y - y₂)$
  $\Rightarrow y₁ + y₂ = -2p$, and $y₁y₂ = -q²$.

The circle with AB as diameter has the equation:

$(x - x₁)(x - x₂) + (y - y₁)(y - y₂) = 0$.

Substitute the factorized forms:

$[x^2 + 2ax - b^2] + [y^2 + 2py - q^2] = 0$
$\Rightarrow x^2 + y^2 + 2ax + 2py - (b^2 + q^2) = 0$.

Thus, the correct option is Option D.