First slide

Analysis of two circles in circles

Question

Circles are drawn on the chords of rectangular hyperbola xy = 4 parallel to line y = x as diameters. All such circles pass through two fixed points whose coordinates are

Difficult

A

$(2, 2)$
B

$(2 \sqrt{2}, 2 \sqrt{2})$
C

$(- 2 \sqrt{2}, - 2 \sqrt{2})$
D

$(2, - 2)$

Solution

$Circles are drawn.........$
$The circle with points (2 t_{1}, 2 / t_{1}) and (2 t_{2}, 2 / t_{2}) as diameter is$
$\begin{array}{l} (x - 2 t_{1}) (x - 2 t_{2}) + (y - 2 / t_{1}) (y - 2 / t_{2}) = 0 \\ Also slope - \frac{1}{t_{1} t_{2}} = 1 t_{1} t_{2} = - 1 \\ So equation of circle will become \end{array}$
$\begin{array}{l} (x^{2} + y^{2} - 8) - (t_{1} + t_{2}) (x - y) = 0 \\ which is of form S + λ L = 0 \end{array}$
$\begin{array}{l} so two fixed points are intersection points of (x - y) = 0 \\ and x^{2} + y^{2} = 8 \\ (2, 2) & (- 2, - 2) \end{array}$

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