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After doing many more problems, I encountered 2 more that I struggle on. Here they are:

 

1. For a certain hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)
where \(a>b,\) the angle between the asymptotes is \(60^\circ.\) Find \(\frac{a}{b}.\)

 

2. An ellipse has foci at \(F_1 = (0,2)\) and \(F_2 = (3,0).\) The ellipse intersects the \(x\)-axis at the origin, and one other point. What is the other point of intersection?

 

Once again, thank you so much for your effort. You do not need to answer both of the question. Any help would be appreciated. UwU

 Jan 29, 2020
 #1
avatar+18250 
+1

For #1:

 

Since a > b, it is longer (in the x-direction) than it is tall (in the y-direction.

 

If you draw the triangle that starts at the center of the hyperbola (0,0) to the right to the point (a,0) and then up to the point (a,b), you will be drawing a right triangle with the angle at the center of the hyperbola having 30°.

 

Therefore, a = sqrt(3) and b = 1.

 Jan 29, 2020
 #2
avatar+194 
+1

That's a smart way to think about it. Thank you for the help, geno!!! wink

 Jan 29, 2020

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