Question

Angle between tangents to the curve y=x²-5x+6 at the points (2, 0) and (3, 0) is:

• A. 0°
• B. 45°
• C. 60°
• D. 90°

Answer 1

The Correct Option is D

To find the angle between the tangents to the curve \( y = x^2 - 5x + 6 \) at the points (2, 0) and (3, 0), we need to determine the derivatives at these points, calculate the slopes of the tangents, and then find the angle between them.

The derivative of the curve is given by:

\( \frac{dy}{dx} = 2x - 5 \)

Calculating the slope at (2, 0):

\( m_1 = 2(2) - 5 = 4 - 5 = -1 \)

Calculating the slope at (3, 0):

\( m_2 = 2(3) - 5 = 6 - 5 = 1 \)

The formula for the angle \( \theta \) between two lines with slopes \( m_1 \) and \( m_2 \) is:

\( \tan \theta = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right| \)

Substituting \( m_1 = -1 \) and \( m_2 = 1 \):

\( \tan \theta = \left|\frac{-1 - 1}{1 + (-1)(1)}\right| = \left|\frac{-2}{1-1}\right| \)

This results in division by zero, indicating \( \tan \theta \) is not defined, which occurs when \( \theta = 90^\circ \).

Thus, the angle between the tangents is \( 90^\circ \).