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Question

Find the points on the curve y = 3x2 − 9x + 8 at which the tangents are equally inclined with the axes.

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Solution

Let (x1, y1) be the required point.
It is given that the tangent at this point is equally inclined to the axes. It means that the angle made by the tangent with the x-axis is ±45°.
∴ Slope of the tangent = tan (±45) = ± 1 ...(1)

Since, the point lies on the curve.Hence, y1=3x12-9x1+8 Now, y=3x2-9x+8dydx=6x-9Slope of the tangent at x1, y1=dydxx1, y1=6x1-9 ...(2)From eq. (1) and eq. (2), we get6x1-9=±16x1-9=1 or 6x1-9=-16x1=10 or 6x1=8x1=106=53 or x1=86=43Also,y1=3532-953+8 or y1=3432-943+8y1=253-453+8 or y1=163-363+8y1=43 or y1=43Thus, the required points are 53, 43 and 43, 43.

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