First slide

Definition of a circle

Question

Extremities of a diagonal of a rectangle are ( 0, 0) and ( 4, 3). The equations of the tangents to the circumcircle of the rectangle which are parallel to the diagonal, are

Moderate

A

$16 x + 8 y \pm 25 = 0$
B

$6 x - 8 y \pm 25 = 0$
C

$8 x + 6 y \pm 25 = 0$
D

none of these

Solution

The equation of the circumcircle of the rectangle is $x (x - 4) + y (y - 3) = 0$
$\Rightarrow x^{2} + y^{2} - 4 x - 3 y = 0 \Rightarrow (x - 2)^{2} + {(y - \frac{3}{2})}^{2} = {(\frac{5}{2})}^{2}$
The equations of the tangents to this circle which are parallel to the diagonal joining (0, 0) and (4, 3) are
$y - \frac{3}{2} = \frac{3}{4} (x - 2) \pm \frac{5}{2} \sqrt{1 + \frac{9}{16}}$ $[\begin{matrix} ∵ Slope of the \\ tangent = 3 / 4 \end{matrix}]$
i.e. $6 x - 8 y + 25 = 0$

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