Points in common

Question

I have the following problem:

How many points do the graphs of $4x^2-9y^2=36$ and $x^2-2x+y^2=15$ have in common?

I know that the answer is in the system of two equations, but how should I solve it?

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Another question: when I try to set the condition of tangence between an hyperbola and an ellipse, I should state $\Delta=0$ and solve for the parameter. The problem is: I need a solving equation, on which I calculate delta.

If I substitute $x^2$ I get a solving equation, if I substitute $y^2$ I get another one. How should I go?

Answer 1

Hint:

Think geometrically - what does the two equations tell you?

Answer 2

From the first equation we have: $$y^2=\frac{4x^2-36}{9},$$ then we substitute $y^2$ in the second equation we found $$\frac{13}{9}x^2-2x-4=0,$$ so we solve for $x$ and then we determine the suitable $y$.