Question

The tangent to the parabola \(x^{2}=2y\) at point P makes an angle of \(45^{\circ }\) with the X-axis, then the coordinates of P are:

• A. \(\left(1,\frac{1}{2}\right)\)
• B. \((2,2)\)
• C. \((-2,2)\)
• D. \(\left(-1,\frac{1}{2}\right)\)

Hint:

The slope of the tangent equals tan 45 degrees, which is 1.

Answer 1

The Correct Option is A

Step 1: Check each option
The slope at $(x_1,y_1)$ equals $x_1$, and it must be 1.

Step 2: Eliminate
Only $x_1=1$ is possible, which leaves the first option.

Step 3: Check
$1^2=2\cdot\dfrac12$, so the point lies on the parabola.

Final Answer:
Only (1, 1/2) lies on the curve and has slope 1. \[ \boxed{\text{(A) }\left(1,\dfrac{1}{2}\right)} \]