Question

Let a > 0, b > 0. Let e and l respectively be the eccentricity and length of the latus rectum of the hyperbola `x^2/"a"^2 - "y"^2/"b"^2` = 1. Let e' and l' respectively the eccentricity and length of the latus rectum of its conjugate hyperbola. If e² = `11/14"l'"` and (e')² = `11/8"l"^'` then the value of 77a + 44b is equal to ______.

Options

• 100

• 110

• 120

• 130

Answer 1

Let a > 0, b > 0. Let e and l respectively be the eccentricity and length of the latus rectum of the hyperbola `x^2/"a"^2 - "y"^2/"b"^2` = 1. Let e' and l' respectively the eccentricity and length of the latus rectum of its conjugate hyperbola. If e² = `11/14"l"` and (e')² = `11/8"l"^'` then the value of 77a + 44b is equal to 130.

Explanation:

∵ We know that eccentricity of the hyperbola and tum of the hyperbola latus rectum of the hyperbola

`x^2/"a"^2 - "y"^2/"b"^2` = 1 are

e = `sqrt(1 + "b"^2/"a"^2)`, ℓ = `(2"b"^2)/"a"`

Now we given that

e² = `11/4l`

⇒ `1 + "b"^2/"a"^2` = `11/14 . (2"b"^2)/"a"`

⇒ `("a"^2 + "b"^2)/"a"^2` = `11/7 . "b"^2/"a"`  ...(i)

And eccentricity and latus rectum of the conjugate hyperbola are

e' = `sqrt(1 + "a"^2/"b"^2)`, ℓ' = `(2"a"^2)/"b"`  ...(A)

And we also given that

(e')² = `11/8ℓ^'`  ...(B)

Now from equations (A) and (B)

⇒ `1 + "a"^2/"b"^2` = `11/8 . (2"a"^2)/"b"`; `("a"^2 + "b"^2)/"b"^2` = `11/4 . "a"^2/"b"`  ...(ii)

Now Equation (i) + Equation (ii)

⇒ `"b"^2/"a"^2` = `4/7 . "b"^3/"a"^3`

∴ 7a = 4b  ...(iii)

Now from (ii)

`((16"b"^2)/49 + "b"^2)/"b"^2` = `11/4 . (16"b"^2)/(49"b")`

`65/49` = `11/4 . 16/49 . "b"`

∴  b = `(4 xx 65)/(11 xx 16)`  ...(iv)

Now we have to find a value of 77a + 44b

11(7a + 4b) = 11(4b + 4b) = 11 × 8b

∴ Value of 11 × 8b = `11 xx 8 xx (4 xx 65)/(16 xx 11)` = 130