First slide

Definition of a circle

Question

The tangents drawn from the origin to the circle
$x^{2} + y^{2} - 2 r x - 2 h y + h^{2} = 0$ are perpendicular, if

Moderate

A

$h = \pm r$
B

$h = \pm 2 r$
C

$h^{2} + r^{2} = 1$
D

$h = \pm 3 r$

Solution

The combined equation of the tangents drawn from $(0, 0)$ to $x^{2} + y^{2} - 2 r x - 2 h y + h^{2} = 0$ , is
$(x^{2} + y^{2} - 2 r x - 2 h y + h^{2}) h^{2} = {(- r x - h y + h^{2})}^{2}$
This equation represents a pair of perpendicular straight lines if
$Coeff. of x^{2} + Coeff. of y^{2} = 0$
$\Rightarrow 2 h^{2} - r^{2} - h^{2} = 0 \Rightarrow r^{2} = h^{2} \Rightarrow r = \pm h$

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