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If a chord joining P(a sec theta, a tan theta), Q(a sec alpha, a tan alpha) on the hyperbola x^(2)-y^(2) =a^(2) is the normal at P, then tan alpha = |
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Answer» <P>`tan theta (4 sec^(2) theta+1)` `:. (tan ALPHA - tan theta)/(sec alpha - sec theta) =- (tan theta)/(sec theta)` `:. tan alpha - tan alpha =- k tantheta` and `sec alpha - sec theta = k sec theta (1+k) sec theta = sec alpha` (1) `:. (1-k) tan theta = tan alpha` (2) `[(1+k)sec theta]^(2) - [(1-k)tan theta]^(2) = sec^(2) alpha - tan^(2) alpha =1` `RARR k =- 2 (sec^(2) theta + tan^(2) theta) =- 4 sec^(2) theta +2` From (2), `tan alpha = tan theta (1+4 sec^(2) theta -2) = tan theta (4 sec^(2) theta -1)` |
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