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

Given that the chord AB of the curve is parallel to the tangent at point \( (4, f(4)) \), the slope of the chord must equal the derivative at \( x = 4 \). We know \( f'(4) = -\frac{3}{4} \).

The slope of the chord AB is calculated using the points \( A(0, \alpha) \) and \( B(8, \beta) \):

\(\text{slope of AB} = \frac{\beta - \alpha}{8 - 0} = \frac{\beta - \alpha}{8}\).

Given this is parallel to the tangent at point \( (4, f(4)) \):

\[\frac{\beta - \alpha}{8} = -\frac{3}{4}\]

Using the information \( f(0) = 2 \), we have \(\alpha = 2\).

Thus, \(\frac{\beta - 2}{8} = -\frac{3}{4}\).

To solve for \(\beta\), multiply both sides by 8:

\[\beta - 2 = -6\]

Adding 2 to both sides gives:

\(\beta = -4\).

Therefore, the correct 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 \( -4 \).