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Question

# Find the intersecting angle between the curves y=x2 and y=x3.

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Solution

## To find the angle of intersection, we first find the point of intersection and then find the angle between the tangents at this point.Hence, the point of intersection of y=x2 and y=x3 can be foud by equating them.∴x2=x3∴x3−x2=0∴x=0 or x=1Hence, the points of intersection are (0,0) and (1,1).The slope of tangent of y=x2 is 2x and the slope of tangent of y=x3 is 3x2 for a given point (x,y) on the curve.Now, at (0,0),slope of tangent of y=x2 = 0slope of tangent of y=x3 = 0As the slopes are same, the angle of intersection is 0∘.Now, at (1,1),slope of tangent of y=x2 = 2slope of tangent of y=x3 = 3therefore angle of intersection can be found by∴tanθ=∣∣∣m2−m11+m1m2∣∣∣∴tanθ=∣∣∣3−21+3×2∣∣∣∴tanθ=∣∣∣17∣∣∣∴θ=tan−117Using log tables, we haveθ=8.13∘Hence the angles of intersection between y=x2 and y=x3 are 0∘ and 8.13∘ at (0,0) and (1,1) respectively.

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