First slide

Hyperbola in conic sections

Question

$The equation 16 x^{2} - 3 y^{2} - 32 x + 12 y - 44 = 0 represents a hyperbola$

Moderate

A

$the length of whose transverse axis is 4 \sqrt{3}$
B

$the length of whose conjugate axis is 4$
C

$whose center is (- 1, 2)$
D

$whose eccentricity is \sqrt{19 / 3}$

Solution

$We have 16 (x^{2} - 2 x) - 3 (y^{2} - 4 y) = 44$
$or 16 (x - 1)^{2} - 3 (y - 2)^{2} = 48$
$or \frac{(x - 1)^{2}}{3} - \frac{(y - 2)^{2}}{16} = 1$
This equation represents a hyperbola with eccentricity
$e = \sqrt{1 + \frac{16}{3}} = \sqrt{\frac{19}{3}}$

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