Question

The tangent to the parabola, x² = 2y at the point \((1,\frac{1}{2})\) makes with the x-axis an angle of :

• A. 0°
• B. 45°
• C. 30°
• D. 60°

Answer 1

The Correct Option is B

First, find the derivative of the parabola equation \(x^2 = 2y\). To do this, implicitly differentiate both sides with respect to \(x\):

\(\frac{d}{dx}(x^2) = \frac{d}{dx}(2y)\)

This yields:

\(2x = 2 \cdot \frac{dy}{dx}\)

Solving for \(\frac{dy}{dx}\), we have:

\(\frac{dy}{dx} = x\)

Next, find the slope of the tangent at the point \((1, \frac{1}{2})\) by substituting \(x = 1\) into the derivative:

\(\frac{dy}{dx} = 1\)

The slope of the tangent line is thus 1.

The tangent of the angle \(\theta\) that the tangent line makes with the x-axis is given by the slope of the tangent line, which is:

\(\tan\theta = 1\)

Thus, \(\theta = \tan^{-1}(1) = 45^\circ\).

Therefore, the solution is that the tangent makes an angle of 45° with the x-axis.