Question

The curvature of the straight line \( y = 2x + 3 \) at \( (1, 5) \) is:

• A. 2
• B. 0
• C. \( \frac{1}{2} \)
• D. 3

Hint:

The curvature of a straight line is always zero because the slope is constant and there is no change in the direction of the line.

Answer 1

The Correct Option is B

The curvature \( \kappa \) of a curve at any point is given by the formula: \[ \kappa = \frac{|y''|}{(1 + (y')^2)^{3/2}} \] where \( y' \) is the first derivative of the curve and \( y'' \) is the second derivative of the curve.
For a straight line, the second derivative \( y'' \) is always 0 because the slope of the line does not change. Therefore, the curvature of any straight line is 0.
For the given straight line \( y = 2x + 3 \), the first derivative \( y' \) (the slope) is 2, and the second derivative \( y'' \) is 0.
Thus, the curvature of the straight line at any point is: \[ \kappa = \frac{0}{(1 + 2^2)^{3/2}} = 0 \] Therefore, the curvature at point \( (1, 5) \) is 0.