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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?

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?

Grade:12

1 Answers

Faiz
107 Points
7 years ago
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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