Problem 170 - A Complete Resource Book in Mathematics for JEE Main

Question

Angle between the tangents to the curve \(y=x^{2}-\) \(5 x+6\) at the points \((2,0)\) and \((3,0)\) is [2006] (A) \(\frac{\pi}{2}\) (B) \(\frac{\pi}{2}\) (C) \(\frac{\pi}{6}\) (D) \(\frac{\pi}{4}\)

Step-by-Step Solution

Answer

The angle between the tangents is \( \frac{\pi}{2} \) radians.

Step 1: Find Derivative of the Curve

The given curve is \( y = x^2 - 5x + 6 \). To find the angle between tangents, we first need to find the derivative of the curve, which gives us the slope of the tangent at any point. The derivative of \( y \) with respect to \( x \) is \( \frac{dy}{dx} = 2x - 5 \).

Step 2: Calculate the Slopes of the Tangents

For the point \((2, 0)\), substitute \( x = 2 \) into the derivative to find the slope: \( m_1 = 2(2) - 5 = -1 \). For the point \((3, 0)\), substitute \( x = 3 \) into the derivative to find the slope: \( m_2 = 2(3) - 5 = 1 \). So, the slopes of the tangents at the points are \( m_1 = -1 \) and \( m_2 = 1 \).

Step 3: Use the Formula for Angle Between Two Lines

The angle \( \theta \) between two lines with slopes \( m_1 \) and \( m_2 \) can be found using the formula: \[ \tan(\theta) = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]Substituting \( m_1 = -1 \) and \( m_2 = 1 \) into the formula gives:\[ \tan(\theta) = \left| \frac{-1 - 1}{1 + (-1)(1)} \right| = \left| \frac{-2}{0} \right| \] which simplifies to an undefined slope, indicating that \( \theta = \frac{\pi}{2} \) radians.

Key Concepts

Derivative of a Curve

A derivative is a fundamental concept in calculus. It gives us the rate at which a function changes at any given point. In the context of a curve, the derivative tells us the slope of the curve at a particular point.

For the curve given by the equation \( y = x^2 - 5x + 6 \), the derivative is computed by differentiating with respect to \( x \). This involves applying power rules of differentiation to each term.

• The derivative of \( x^2 \) is \( 2x \).
• The derivative of \(-5x\) is \(-5 \).
• The derivative of a constant, such as 6, is 0.

Thus, the derivative \( \frac{dy}{dx} = 2x - 5 \) represents the slope of the tangent to the curve at any point \( (x, y) \). This derivative is crucial for finding the slopes at specific points on the curve.

Slope of Tangent

The slope of a tangent to a curve at a particular point indicates how steep the line is at that point. It also tells you whether the line is increasing or decreasing.

Using the derivative \( \frac{dy}{dx} = 2x - 5 \) found in the previous section, we calculate the slopes at specific points by substituting the \( x \)-values of these points into the derivative.

• At point \((2, 0)\), substitute \( x = 2 \) into the derivative to get \( m_1 = 2(2) - 5 = -1 \). The negative slope indicates that the tangent is decreasing at this point.
• At point \((3, 0)\), substitute \( x = 3 \) into the derivative to get \( m_2 = 2(3) - 5 = 1 \). The positive slope shows that the tangent is increasing at this point.

By knowing these slopes, we can further compute the angle between these tangents.

Angle Between Lines Formula

The angle between two lines can be determined using their slopes. When two tangents intersect, the slope values help calculate the angle between them. These pairs of slopes, \( m_1 \) and \( m_2 \), are used in the formula:\[ \tan(\theta) = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]This formula is derived from the tangent of the angle between lines and allows you to determine the angle \( \theta \) in radians.

• By substituting \( m_1 = -1 \) and \( m_2 = 1 \) into the formula, we see:
  \[ \tan(\theta) = \left| \frac{-1 - 1}{1 + (-1)(1)} \right| = \left| \frac{-2}{0} \right| \]
• The result is an undefined slope, which mathematically implies that \( \theta = \frac{\pi}{2} \). This means the tangents are perpendicular to each other.

Understanding this relationship helps to visualize how the lines interact and meet each other in the Cartesian plane.