Show that the following set of curves intersect orthogonally: (i) `y=x^3 and 6y=7-x^2`, (ii) `x^3-3x y^2=-2 and 3x^2y-y^3=2.` (iii) `x^2+4y^2=8 and x^2-2y^2=4`

Explanation

To show that the given set of curves intersect orthogonally, we need to find the points of intersection of the curves and then calculate the slopes of their tangents at those points. If the product of the slopes is -1, then the curves intersect orthogonally.

### Part (i): Curves \(y = x^3\) and \(6y = 7 - x^2\)

1. **Find the intersection points:**
- Substitute \(y = x^3\) into \(6y = 7 - x^2\):
\[
6(x^3) = 7 - x^2 \implies 6x^3 + x^2 - 7 = 0
...