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Tamil Nadu Board of Secondary EducationHSC Science Class 12

Prove that the length of the latus rectum of the hyperbola abx2a2-y2b2 = 1 is ba2b2a - Mathematics

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Question

Prove that the length of the latus rectum of the hyperbola `x^2/"a"^2 - y^2/"b"^2` = 1 is `(2"b"^2)/"a"`

Sum
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Solution

The latus rectum LL’ of an hyperbola `x^2/"a"^2 - y^2/"b"^2` = 1 passes through S(ae, 0)

Hence L is (ae, y1)

`("a"^2"e"^2)/"a"^2 - y_1^2/"b"^2` = 1

`"e"^2 - 1 = y_1^2/"b"^2`

`y_1^2 = "b"^2("e"^2 - 1)`

= `"b"^2(1 + "b"^2/"a"^2 - 1) (because "e"^2 = 1 + "b"^2/"a"^2)`

`y_1^2 = "b"^4/"a"^2`

`y_1 = +-  "b"^2/"a"`

End points of latus rectums are `("ae", "b"^2/"a")` and `("ae", - "b"^2/"a")`

∴ LL' = `"b"^2/"a" + "b"^2/"a"`

LL' = `(2"b"^2)/"a"`

Hence proved.

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Chapter 5: Two Dimensional Analytical Geometry-II - Exercise 5.2 [Page 197]

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Samacheer Kalvi Mathematics - Volume 1 and 2 [English] Class 12 TN Board
Chapter 5 Two Dimensional Analytical Geometry-II
Exercise 5.2 | Q 6 | Page 197

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