Question

f(x) and g(x) intersect at (3,5). If given that f’(3) = 7 and g’(3) = -1 then find the angle of intersection between f(x) and g(x) at (3,5)

• A. θ = tan^-1 (4 / 3)
• B. θ = tan^-1 (3 / 4)
• C. θ = tan^-1 (5 / 3)
• D. None of these

Answer 1

The correct option is A

θ = tan^-1 (4 / 3)

The angle between two curves at the point of intersection is defined as the angle between the tangents drawn at that point on the curves.

We know that if the slope of two tangent are $m_1$ ,$m_2$ at a point (h,k) then the angle between them will be given by

tan$\theta =\left|\frac{m_1-m_2}{1+m_1,m_2}\right|$

We know that slope of the tangent is nothing but first derivative of f(x) or f’(x) at that point.

$m_1$ = f'(3) = 7

$m_2$ = g'(3) = -1

So, tan$\left(\theta \right)$ = $\left|\frac{7-(-1)}{1+\left(7\right)(-1)}\right|$

Or tan $\left(\theta \right)$ = 4/3

Or $\theta$ = $ta{n}^{-1}$ (4/3)