prove that the distance between ( at 2 , 2at ) and ( a/ t2 , - 2a/ t - Maths - Coordinate Geometry

Question

prove that the distance between ( at 2 , 2at ) and (
a/ t2 , - 2a/ t ) is a( t+ 1/t)2

Lalit Mehra · Accepted Answer

Let the given point be A(at2, 2at) and $B (\frac{a}{t^{2}}, \frac{- 2 a}{t})$ Distance between the given point, AB $= \sqrt{{(a t^{2} - \frac{a}{t^{2}})}^{2} + [2 a t - (- \frac{2 a}{t})]^{2}} = \sqrt{a^{2} (t^{2} - \frac{1}{t^{2}})^{2} + [2 a (t + \frac{1}{t})]^{2}} = \sqrt{a^{2} (t^{4} + \frac{1}{t^{4}} - 2) + 4 a^{2} (t^{2} + \frac{1}{t^{2}} + 2)} = a \sqrt{t^{4} + \frac{1}{t^{4}} - 2 + 4 t^{2} + \frac{4}{t^{2}} + 8} = a \sqrt{t^{4} + 4 t^{2} + 6 + \frac{4}{t^{2}} + \frac{1}{t^{4}}} = a \sqrt{(t + \frac{1}{t})^{4}} [∵ (x + y)^{4} = x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3} + y^{4}] = a \sqrt{[(t + \frac{1}{t})^{2}]^{2}} = a (t + \frac{1}{t})^{2}$

prove that the distance between ( at 2 , 2at ) and ( a/ t2 , - 2a/ t ) is a( t+ 1/t)2

prove that the distance between ( at ² , 2at ) and ( a/ t² , - 2a/ t ) is a( t+ 1/t)²