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QuestionFind points A, B, and C on y=36x2y=\frac{\sqrt{3}}{6} x^{2} such that ∠OPA=∠OPB=30∘\angle OPA = \angle OPB = 30^{\circ} and ∠ABC=60∘\angle ABC = 60^{\circ}.

Studdy Solution

STEP 1

Assumptions1. The origin O is at20. Point is at (−3,0)\left(-\frac{3}{},0\right)3. There are three points A, B, and C on the graph of the function y=36xy=\frac{\sqrt{3}}{6} x^{}
4. ∠OPA=∠OPB=30∘\angle OPA = \angle OPB =30^{\circ}
5. ∠ABC=60∘\angle ABC =60^{\circ}

STEP 2

We know that the equation of the line OP is y=0y =0 since lies on the x-axis.

STEP 3

The equation of the line OA can be found by using the fact that ∠OPA=30∘\angle OPA =30^{\circ}. The slope of line OA is tan⁡(30∘)=13\tan(30^{\circ}) = \frac{1}{\sqrt{3}}.

STEP 4

Since the line OA passes through the origin O(20,0), the equation of the line OA is y=13(x−20)y = \frac{1}{\sqrt{3}}(x -20).

STEP 5

Similarly, the equation of the line OB can be found by using the fact that ∠OPB=30∘\angle OPB =30^{\circ}. The slope of line OB is tan⁡(30∘)=13\tan(30^{\circ}) = \frac{1}{\sqrt{3}}.

STEP 6

Since the line OB passes through the origin O(20,0), the equation of the line OB is y=13(x−20)y = \frac{1}{\sqrt{3}}(x -20).

STEP 7

The coordinates of points A and B can be found by solving the equations of lines OA and OB with the equation of the parabola y=36x2y=\frac{\sqrt{3}}{6} x^{2}.

STEP 8

olving the equation of line OA with the equation of the parabola gives13(x−20)=36x2\frac{1}{\sqrt{3}}(x -20) = \frac{\sqrt{3}}{6} x^{2}

STEP 9

olving this equation gives the x-coordinates of points A and B.

STEP 10

Substitute the x-coordinates of points A and B into the equation of the parabola to get their y-coordinates.

STEP 11

The coordinates of point C can be found by using the fact that ∠ABC=60∘\angle ABC =60^{\circ}. The slope of line BC is tan⁡(60∘)=3\tan(60^{\circ}) = \sqrt{3}.

STEP 12

Since the line BC passes through point B, the equation of the line BC is y−yB=(x−xB)y - y_{B} = \sqrt{}(x - x_{B}).

STEP 13

olving the equation of line BC with the equation of the parabola gives the coordinates of point C.

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