Question 14

Let the foci of a hyperbola be $$(1, 4)$$ and $$(1, -12).$$ If it passes through the point $$(1, 6)$$, then the length of its latus-rectum is :

$$S(1, 14), \quad S'(1, -12), \quad P(1, 6)$$

$$SP = \vert{}14 - 6\vert{} = 8$$

$$S'P = \vert{}6 - (-12)\vert{} = 18$$

$$2b = \vert{}SP - S'P\vert{} = \vert{}8 - 18\vert{} = 10 \implies b = 5$$

$$2be = SS' = 14 - (-12) = 26 \implies be = 13$$

$$a^2 = b^2e^2 - b^2 = (be)^2 - b^2 = 13^2 - 5^2 = 169 - 25 = 144$$

$$\text{L.R.} = \frac{2a^2}{b} = \frac{2(144)}{5} = \frac{288}{5}$$

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