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Solution to the problem of the (previous) week
Problem:The figure shows the parabola , that is tangent to the hyperbola at and intersects it at .
Find the parameter as well as the point of tangency and the point of intersection .
(The results should be correct to four decimal places.)
Solution:Equating the slopes at the point of tangency, we obtain
Hence, we have
Simplifying, we get and , which yields
Substitute in the equation of the parabola to find the intersection point:
Since is a zero of order 2 of (-coordinate of the point of tangency), we can factor and obtain the form