The tangent to the parabola, \(x^2 = 2y\) at the point \(\left(1, \displaystyle\frac{1}{2}\right)\) makes with the \(x\)-axis an angle of:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 45°
The slope of the tangent at a point is the value of dy/dx there, and the slope equals \(\tan\theta\), where \(\theta\) is the angle the tangent makes with the x-axis.
Differentiating \(x^2 = 2y\) with respect to x:
\( \displaystyle 2x = 2\frac{dy}{dx} \)
\( \displaystyle \frac{dy}{dx} = x \)
At \(\left(1, \frac{1}{2}\right)\):
\(\tan\theta = 1\)
\(\theta = 45^{\circ}\)