Let a > 0, b > 0. Let e and l respectively be the eccentricity and length of the latus rectum of the hyperbola x²/a² - y²/b² = 1. Let e' and l' respectively be the eccentricity and length of the latus rectum of its conjugate hyperbola. If e² = 11/14l and ( e' )² = 11/8l', then the value of 77a + 44b is equal to :

Solution

Maths sir · 4 steps

Write both conditions in a and b, then divide one by the other: the ugly a²+b² cancels.

1. Standard results. For x²/a²-y²/b²=1: e² = (a²+b²)/a² and l = 2b²/a. For the conjugate hyperbola the roles of a and b swap, so e'² = (a²+b²)/b² and l' = 2a²/b.
2. Two equations. e² = 11/14·2b²/a gives 7(a²+b²) = 11ab². Likewise, e'² = 11/8·2a²/b gives 4(a²+b²) = 11a²b.

   7(a²+b²)/4(a²+b²) = 11ab²/11a²b ⇒ 7/4 = b/a ⇒ b = 7a/4

3. Find a and b. Put b = 7a/4 into 4(a²+b²) = 11a²b: 4 × 65a²/16 = 77a³/4, so 65a² = 77a³ and a = 65/77. Then b = 7/4×65/77 = 65/44.
4. So. 77a + 44b = 65 + 65 = 130, which is option (d).

The idea to keep. A hyperbola and its conjugate share a²+b². When you get one condition on each, divide the two equations: that common factor cancels and you are left with a simple ratio of a and b.

Answer from the NTA final answer key. Solution by PrepWiser.