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Problem 143

AMC 10/12, early questions

Geometry Difficulty 3.5 Find the answer

Given that point $A$ is on the parabola $C$ : $x^{2}=2py (p > 0)$ , and $O$ is the coordinate origin. If $A$ and $B$ are the two common points of the circle with center $M(0,10)$ and radius $|OA|$ , and $\triangle ABO$ is an equilateral triangle, then the value of $p$ is $\_\_\_\_\_\_$ .

Your answer

Official solution

By the symmetry of the parabola, points $A$ and $B$ are symmetric about the $y$ -axis. Without loss of generality, let $A(a,b)$ be in the first quadrant.

Since $\triangle ABO$ is an equilateral triangle, $\angle AOM=30^{\circ}$ .

Given $|MA|=|MO|=10$ , we have $\angle OMA=120^{\circ}$ .

Thus, $|OA|= \sqrt {|OM|^{2}+|MA|^{2}-2|OM|\cdot |MA|\cos 120 ^{\circ} }=10 \sqrt {3}$ .

Hence, $A(5 \sqrt {3},15)$ .

Substitute the coordinates of $A$ into the equation of the parabola: $(5 \sqrt {3})^{2}=2p \times 15$ , solving for $p$ gives $p= \dfrac {5}{2}$ .

Therefore, the answer is: $\boxed{ \dfrac {5}{2}}$ .

This problem involves using properties of equilateral triangles and parabolas to find the coordinates of point $A$ , and then substituting these values into the equation of the parabola to find the value of $p$ . This is a basic problem that tests your understanding of these concepts.

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