Let y=f(x) and z=g(x) be two curves. Then the tangents drawn to curves at the intersection of these curve is perpendicular to each other if and only if f'(x).g'(x)=-1

Explanation

To prove that the tangents drawn to the curves \( y = f(x) \) and \( z = g(x) \) at their intersection points are perpendicular if and only if \( f'(x) \cdot g'(x) = -1 \), we will follow these steps:

### Step 1: Identify the intersection point
Let the curves intersect at the point \( (m, f(m)) \) where \( f(m) = g(m) \).

### Step 2: Find the slopes of the tangents
The slope of the tangent to the curve \( y = f(x) \) at \( x = m \) is given by the derivative \( f'(m) \).
Similarly, the slope of the tangent to the curve \( z = g(x) \) at \( x = m \) is given by the derivative \( g'(m) \).
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