Question

Angle between the tangents to the curve $y=x^2-5x+6$ at points $(2,0)\&(3,0)$is

• A. $\frac{\mathrm{\pi }}{3}$
• B. $\frac{\mathrm{\pi }}{2}$
• C. $\frac{\mathrm{\pi }}{6}$
• D. $\frac{\mathrm{\pi }}{4}$

Answer 1

The correct option is B

$\frac{\mathrm{\pi }}{2}$

Explanation for the correct option:

Step 1. Differentiate the given equation

Given equation of curve

$y=x^2-5x+6$

$\Rightarrow \frac{dy}{dx}=2x-5$

Step 2. Find slope of tangent at point $\mathbf{(}\mathbf{2}\mathbf{,}\mathbf{}\mathbf{0}\mathbf{)}$

$\begin{array}{rcl}& \Rightarrow & m_1={\frac{dy}{dx}}_{(2,0)}\\ & =& 2\times 2-5\\ & =& -1\end{array}$

Step 3. Find slope of tangent at point $\mathbf{(}\mathbf{3}\mathbf{,}\mathbf{}\mathbf{0}\mathbf{)}$

$\begin{array}{rcl}& \Rightarrow & m_2={\frac{dy}{dx}}_{(3,0)}\\ & =& 2\times 3-5\\ & =& 1\end{array}$

Since, ${\mathit{m}}_{\mathbf{1}}\mathbf{\times }{\mathit{m}}_{\mathbf{2}}\mathbf{=}\mathbf{-}\mathbf{1}$ the two tangents are perpendicular to each other.

Therefore the angle between the tangents is $\frac{\mathbf{\pi }}{\mathbf{2}}$

Hence, the correct option is $\mathbf{\left(}\mathit{B}\mathbf{\right)}$.