Q.Show that the curves $y^2 = 4(x + 1)$ and $y^2 = 36(9 - x)$ intersect orthogonally.

Angle Between Two Curves

The angle between two curves at a point where they meet is defined as the angle between their tangent lines at that common point. Since the slope of a tangent is the derivative, we first find the point of intersection, then compute each curve's slope there using $\frac{dy}{dx}$. If the two slopes are $m_1$ and $m_2$, the acute angle $\theta$ between the curves satisfies

The procedure is: solve the two equations simultaneously to get the intersection point(s); differentiate each curve and evaluate at that point to obtain $m_1$ and $m_2$; then apply the formula. For example, if at a shared point one tangent has slope $m_1 = 2$ and the other $m_2 = -\tfrac13$, then $\tan\theta = \left|\frac{2 + 1/3}{1 - 2/3}\right| = \left|\frac{7/3}{1/3}\right| = 7$, so $\theta = \tan^{-1}7$. …