Difference between revisions of "2006 SMT/Team Problems/Problem 13"
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+ | A ray is drawn from the origin tangent to the graph of the upper part of the hyperbola <math> y^2=x^2-x+1 </math> in the first quadrant. This ray makes an angle of <math> \theta </math> with the positive <math> x </math> axis. Compute <math> \cos\theta </math>. | ||
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==Solution== | ==Solution== | ||
Latest revision as of 17:38, 14 January 2020
Problem
A ray is drawn from the origin tangent to the graph of the upper part of the hyperbola in the first quadrant. This ray makes an angle of with the positive axis. Compute .
Solution
A line that passes through the origin has an equation of . If the line is tangent to the hyperbola than the equation will have only one solution. This means that the discriminant of the equation will be equal to zero. Solving:
We can ignore the negative root of the equation because the line is tangent to the parabola in the first quadrant. Therefore .
We now need to find the cosine of the angle formed between the line and the x-axis. We can do this by forming a right triangle using an arbitrary point on the line, and the x-axis. We can then solve for the cosine of the angle.
Picking the point , we find that the hypotenuse of the right triangle formed using the x-axis as a side is . Therefore, the cosine of the angle formed between the line and the x-axis is