The asymptotes of a hyperbola are parallel to $2 x + 3 y = 0&3 x + 2 y = 0$ . Its centre is $(1, 2) &$ it passes through $(5, 3)$ . Find the equation of the hyperbola.

If the eqn is (ax + by - c)(dx +ey -f) -154 = 0

find a + b + c + d + e - f

Question

The asymptotes of a hyperbola are parallel to $2 x + 3 y = 0&3 x + 2 y = 0$ . Its centre is $(1, 2) &$ it passes through $(5, 3)$ . Find the equation of the hyperbola.

If the eqn is (ax + by - c)(dx +ey -f) -154 = 0

find a + b + c + d + e - f

TestHub · Accepted Answer

Let the asymptotes be 2 x + 3 y + λ = 0 and 3 x + 2 y + μ = 0 . Since asymtotes passes through ( 1 , 2 ) , then λ = − 8 and μ = − 7 Thus the equation of asymptotes are 2 x + 3 y − 8 = 0 and 3 x + 2 y − 7 = 0 Let the equation of hyperbola be ( 2 x + 3 y − 8 ) ( 3 x + 2 y − 7 ) + v = 0 It passes through ( 5 , 3 ) , then ​ ( 10 + 9 − 8 ) ( 15 + 6 − 7 ) + v = 0 ⇒ ​ 11 × 14 + v = 0 ∴ ​ v = − 154 ​ putting the value of v in (1) we obtain ( 2 x + 3 y − 8 ) ( 3 x + 2 y − 7 ) − 154 = 0 which is the equation of required hyperbola. \end{document}

Mathematics - Hyperbola Question with Solution | TestHub

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