Question

Let \( f(x) \) be a differentiable function, \( A(0, \alpha) \) and \( B(8, \beta) \) be two points on the curve \( y = f(x) \). Given \( f(0) = 2 \) and \( f'(4) = -\frac{3}{4} \). If the chord \( AB \) of the curve is parallel to the tangent drawn at the point \( (4, f(4)) \), then \( \beta \) is:

• A. \( -4 \)
• B. \( -6 \)
• C. \( 2 \)
• D. \( 8 \)

Hint:

To solve such problems, always remember that the slope of the chord is equal to the slope of the tangent if the chord is parallel to the tangent.

Answer 1

The Correct Option is A

To solve the problem, we need to establish that the slope of the chord \( AB \) is equal to the slope of the tangent at point \( (4, f(4)) \).
First, calculate the slope of the tangent at \( (4, f(4)) \):
\[ f'(4) = -\frac{3}{4} \]
This is the given derivative and therefore the slope of the tangent.
Now, calculate the slope of the chord \( AB \):
The slope of a line between two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by:
\[ \text{slope of } AB = \frac{y_2 - y_1}{x_2 - x_1} \]
Substitute the points \( A(0, \alpha) \) and \( B(8, \beta) \):
\[ \text{slope of } AB = \frac{\beta - \alpha}{8 - 0} = \frac{\beta - \alpha}{8} \]
Since the slope of the chord \( AB \) is parallel to the slope of the tangent at \( (4, f(4)) \), we equate the slopes:
\[\frac{\beta - \alpha}{8} = -\frac{3}{4}\]
Given \( f(0) = \alpha = 2 \), substitute this value:
\[\frac{\beta - 2}{8} = -\frac{3}{4}\]
To solve for \( \beta \), multiply both sides by 8:
\[\beta - 2 = -\frac{3}{4} \times 8\]
\[\beta - 2 = -6\]
Add 2 to both sides:
\[\beta = -6 + 2\]
\[\beta = -4\]
Thus, the value of \( \beta \) is \(-4\).

Answer 2

We are given the following conditions: - \( f(0) = 2 \) - \( f'(4) = -\frac{3}{4} \) - The chord \( AB \) is parallel to the tangent at \( (4, f(4)) \). 
Step 1: Slope of the tangent at \( x = 4 \). The slope of the tangent at \( x = 4 \) is given by \( f'(4) \), which is \( -\frac{3}{4} \). 
Step 2: Slope of the chord \( AB \). The slope of the chord \( AB \) is given by the difference in the \( y \)-coordinates of \( A \) and \( B \) divided by the difference in the \( x \)-coordinates of \( A \) and \( B \): \[ \text{slope of } AB = \frac{\beta - \alpha}{8 - 0} = \frac{\beta - \alpha}{8} \] Since the chord \( AB \) is parallel to the tangent at \( (4, f(4)) \), the slope of the chord is equal to the slope of the tangent, i.e., \[ \frac{\beta - \alpha}{8} = -\frac{3}{4} \] 
Step 3: Substitute \( \alpha = f(0) = 2 \). Substituting \( \alpha = 2 \) into the equation: \[ \frac{\beta - 2}{8} = -\frac{3}{4} \] Step 4: Solve for \( \beta \). Multiply both sides by 8 to eliminate the denominator: \[ \beta - 2 = -6 \] Finally, add 2 to both sides: \[ \beta = -4 \] Thus, the value of \( \beta \) is \( -4 \).