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If lines 2x + 3y =10 and 2x-3y =10 are tangents at the extremities of a latus rectum of an ellipse whose centre is origin,then the length of the latus rectum is?
You know both the tangents...Their point of intersection is (5,0)...You can use the formula of pair of tangents from a point drwn from (x`, y`)....that is SS` - T² = 0......S= x²/a² + y²/b² - 1 = 0...S`= x`²/a² + y`²/b² - 1 = 0...T= xx`/a² + yy`/b² - 1 = 0...Here (x`,y`) is (5,0)....Multiply the given tangents and then compare the coefficients....A little simplified form which you will get is:::: x²/a² + y²/b²*(1- 25/a²) - 10x/a² + (25/a²) = 0....And product of tangents is:::: 4x² - 9y² - 40x + 100 = 0.....Now comapre the coefficients...from which you will get a²=1/4 and b²=11...which gives length of the latus ractum as ans:::: 44 units......
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