SAT1000 - P846

Geometry Level pending

As shown above, O N , N M ON, NM are rigid rods and D N = O N = 1 , M N = 3 DN=ON=1,MN=3 , O ( 0 , 0 ) O(0,0) is fixed on the coordinate plane, and D D is restricted along the x-axis. Then as D D moves horizontally, point M M will rotate around point O O . Curve C C is the locus of point M M .

If line l l intersects with l 1 : x 2 y = 0 l_1:x-2y=0 at point P P , l 2 : x + 2 y = 0 l_2:x+2y=0 at point Q Q , and l l is tangent to curve C C .

Then find the minimum area of O P Q \triangle OPQ when line l l moves and rotates.

Let S S be the minimum area. Submit 1000 S \lfloor 1000S \rfloor .


Have a look at my problem set: SAT 1000 problems


The answer is 8000.

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1 solution

Locus of M M , assuming the two rods are hinged at their junction and the rod O N \overline {ON} is hinged at O O , is x 2 + 4 y 2 = 16 x^2+4y^2=16

The triangle O P Q \triangle {OPQ} will have minimum area when P Q \overline {PQ} is perpendicular to the x x -axis, the minimum area being

1 2 × 4 × 4 = 8 \dfrac {1}{2}\times 4\times 4=8 square units, so that the answer is 8000 \boxed {8000}

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