Question

If the tan of angle of intersection of y = $x^2$ and y = $x^3$ in the first quadrant is m, find the value of $\left|\frac{1}{m}\right|$

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Answer 1

y = $x^2$ and y = $x^3$

Angle of intersection between two curves is the angle between the tangents drawn to the curves at the point of intersection. So, we will use following steps to find the angle

1. Find the coordinates of the point where the curves intersect
2. Find the slope of tangents at the point of intersection
3. Find the angle between tangents using slopes as $\tan \theta =\left|\frac{m1-m2}{1+m1\cdot m2}\right|$

1. To find the point of intersection, we equate the x-coordinates

=> x² = x³ => x = 0 or 1

So, there are two points where these curves meet. But we have to find the angle of intersection in first quadrant. This means only x = 1 is valid.

2. Now to find the angle of tangents

a) For y = $x^2$

y'= 2x $\Rightarrow$ m1 = 2

b) For y = $x^3$

y' = 3 $x^2$ $\Rightarrow$ m2 = 3

3. We got m1 = 2 and m2 = 3

Now
$m=\tan \theta =\left|\frac{m1-m2}{1+m1.m2}\right|=\left|\frac{2-3}{1+6}\right|$

$\Rightarrow m=\frac{1}{7}$

$\Rightarrow \left|\frac{1}{m}\right|=7$