Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 3.0 Junior Find the answer

Point $M$ is on the parabola $C:y^{2}= 2px(p > 0)$ . $F$ is the focus of the parabola $C$ , and $D$ is the origin. If $|MF|=p$ , and $K$ is the intersection point of the directrix of the parabola $C$ and the $x$ -axis, then $\angle MKO=$

Pick one

Solution

Analysis

This question examines the equation and definition of a parabola, as well as the calculation of slope, which is quite basic.

Given the conditions, we can select point $M\left( \frac{p}{2},p\right)$ and $K\left(- \frac{p}{2},0\right)$ , from which the conclusion can be directly drawn.

Solution

Given the conditions, we select point $M\left( \frac{p}{2},p\right)$ ,

Since $K\left(- \frac{p}{2},0\right)$ ,

Therefore, $k_{KM}=1$ ,

Hence, $\angle MKO=45^{\circ}$ ,

Therefore, the correct choice is $\boxed{C}$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.