First slide

Hyperbola in conic sections

Question

Equation of the rectangular hyperbola whose focus is (1,-1) and the corresponding directrix is x - y + 1 = 0 is

Moderate

A

$x^{2} - y^{2} = 1$
B

xy = 1
C

2xy - 4x + 4y + 1 =0
D

2xy + 4x - 4y - 1 = 0

Solution

$Eccentricity of rectangular hyperbola is \sqrt{2}$
$∴ Equation of hyperbola is$
$\sqrt{(x - 1)^{2} + (y + 1)^{2}} = \sqrt{2} \frac{| x - y + 1 |}{\sqrt{1^{2} + (- 1)^{2}}}$
$\begin{array}{l} or (x - 1)^{2} + (y + 1)^{2} = (x - y + 1)^{2} \\ or 2 xy - 4 x + 4 y + 1 = 0 \end{array}$

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