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Definition of Circle

L1 is a line ...

Question

$L_{1}$ is a line intersecting $x$ and $y$ axes at $A (a, 0)$ and $B (0, b) .$ $L_{2}$ is a line perpendicular to $L_{1}$ intersecting $x$ and $y$ axes at $C$ and $D$ respectiveley. What is the condition for the common chord of the circles with $B D$ and $A C$ as diameters to pass through the point $(a, b) ?$

a = 2 b

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2 a = b

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a = - b

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a = - 2 b

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Solution

The correct option is C $a = - b$
$B (0, b), D (0, - a α)$
$x^{2} + (y - b) (y + a α) = 0$ (Circle with $B D$ as diameter)
$x^{2} + y^{2} + (a α - b) y - a b α = 0$
$A (a, 0), C (α b, 0)$
( $A C$ as diameter)
$(x - a) (x - α b) + y^{2} = 0 \Rightarrow x^{2} + y^{2} - (a + α b) x + a b α = 0$
Two equations are
$x^{2} + y^{2} + (a α - b) y - a b α = 0$
$x^{2} + y^{2} - (a + α b) x + a b α = 0$
On subtracting the above equations , we get
$(a + α b) x + (a α - b) y - 2 a b α = 0$ (equation of chord)
Since, the chord passes through $(a, b)$ therefore, put $x = a, y = b$
$\Rightarrow a^{2} = b^{2}$
$\Rightarrow a = \pm b$

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L1 is a line intersecting x and y axes at A(a,0) and B(0,b). L2 is a line perpendicular to L1 intersecting x and y axes at C and D respectiveley. What is the condition for the common chord of the circles with BD and AC as diameters to pass through the point (a,b)?