Consider the hyperbola a 2 x 2 − b 2 y 2 = 1 having one of its focus at P ( − 3 , 0 ) . If the latus ractum through its other focus subtends a right angle at P and a 2 b 2 = α 2 − β , α , β ∈ N

Question

Consider the hyperbola a 2 x 2 ​ − b 2 y 2 ​ = 1 having one of its focus at P ( − 3 , 0 ) . If the latus ractum through its other focus subtends a right angle at P and a 2 b 2 = α 2 ​ − β , α , β ∈ N .

TestHub · Accepted Answer

​ f 1 ​ ≡ ( − ae , 0 ) ≡ P ( − 3 , 0 ) ⇒ ae = 3 ​ ​ tan 4 5 ∘ = 2 a e b 2 / a ​ 2 a e = a b 2 ​ b 2 = 6 a Also a 2 e 2 = a 2 + b 2 9 = a 2 + 6 a a 2 + 6 a − 9 = 0 a = − 3 ± 3 2 ​ = − 3 ( 1 ± 2 ​ ) ∴ a 2 b 2 = a 2 ⋅ 6 a = 6 a 3 = 6 ( 135 2 ​ − 189 ) α = 810 and β = 1134 ∴ α + β = 1944 ​

Mathematics - Hyperbola Question with Solution | TestHub

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