Question

The chord joining the points \( (5,5) \) and \( (11,227) \) on the curve \( y = 3x^2 - 11x - 15 \) is parallel to tangent at a point on the curve. The abscissa of the point is

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

Hint:

Parallel lines have equal slopes — use this to link chord and tangent.

Answer 1

The Correct Option is D

Concept: If a chord is parallel to a tangent, then their slopes are equal.

Step 1: Find slope of the chord.
Points: \[ (5,5), \quad (11,227) \] Slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] \[ m = \frac{227 - 5}{11 - 5} \] \[ = \frac{222}{6} = 37 \]

Step 2: Differentiate the curve.
\[ y = 3x^2 - 11x - 15 \] \[ \frac{dy}{dx} = 6x - 11 \]

Step 3: Equate slope of tangent to chord.
\[ 6x - 11 = 37 \]

Step 4: Solve equation.
\[ 6x = 48 \] \[ x = 8 \]

Step 5: Interpretation.
At \( x = 8 \):
• Tangent slope = 37
• Same as chord slope
• Hence parallel

Step 6: Final Answer.
\[ \boxed{8} \]