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For the curve `y=4x^3-2x^5,`find all the points at which the tangent passes through the origin. |
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Answer» Equation of curve ` y = 4x^(3) - 2x^(5)` ….(1) ` rArr (dy)/(dx) = 12x^(2) - 10x^(4)` Slope of tangent at point `(x_(1), y_(1)) = 12x_(1)^(2) - 10x_(1)^(4)` and equation of tangent. ` y - y_(1) = (12x_(1)^(2)-10x_(1)^(4))(x-x_(1))` This tangent passes through the point (0, 0). `:. 0-y_(1)=(12x_(1)^(2)-10x_(1)^(4))(0-x_(1))` `rArr y_(1) = 12x_(1)^(3) - 10x_(1)^(5)` ....(2)` `(x_(1), y_(1))` line on curve (1) ` :. y_(1)= 2x_(1)^(5)` `rArr 12x_(1)^(3)-10x_(1)^(5)=4x_(1)^(3)-2x_(1)^(5)` `rArr 8x_(1)^(3)-8x_(1)^(5)= 0` `rArr 8x_(1)^(3) (1-x_(1)^(2))= 0` ` rArr x_(1) = 0, 1, -1` From equation (2), the corresponding values of `y_(1)` are 0, 2 and -2. `:. " Required point " -=(0, 0), (1, 2) and (-1, 2) and (-1, -2)`. |
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