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

$Let the equation of any normal be y = - t x + 2 t + t^{3} .$
$Since it passes through the point (15, 12), we have$
$\begin{aligned} 12 = - 15 t + 2 t + t^{3} \\ or t^{3} - 13 t - 12 = 0 \end{aligned}$
$One root is - 1 . Then,$
$\begin{aligned} (t + 1) (t^{2} + t - 12) = 0 \\ or t = 1, 3, 4 \end{aligned}$
$Therefore, the co-normal points are (1, - 2), (9, - 6), and (16, 8) .$
$Therefore, the centroid is (26 / 3, 0) .$

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