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Through the vertex O of the parabola y² = 4ax, two chords OP and OQ are drawn. A circle is drawn with OP and OQ as diameters, and it intersects at point R (other than O). If theta1 and theta2 are the angles that the tangents to the parabola at P and Q make with the axis of the parabola, and phi is the angle that OR makes with the axis of the parabola, then find the value of cot(theta1) + cot(theta2) + 2*tan(phi).
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Correct answer: 0
Solution
Parametrically P = (at1², 2at1), Q = (at2², 2at2). The tangent slope at P = 1/t1 so cot(theta1) = t1; similarly cot(theta2) = t2. For the circle on OP, OQ as diameters, OR is the second intersection. Since angle ORP = 90 and angle ORQ = 90 (angles in semicircle), R lies on both circles, meaning OR is perpendicular to PQ. The slope of PQ = 2/(t1+t2), so slope of OR = -(t1+t2)/2, giving tan(phi) = -(t1+t2)/2. Thus cot(theta1) + cot(theta2) + 2*tan(phi) = t1 + t2 + 2*(-(t1+t2)/2) = t1 + t2 - (t1+t2) = 0.
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