Expert: Ahmed Salami - 8/3/2004

Question
Hi. Since my question must be accompanied with the visual XY-plane graph, I'll try to describe it after the question.
Thanks and regards,

Aes

Question:
In the XY-plane, the equation of the circle shown is X squared + Y squared = 4. At how many points does the graph of the equation Y = X squared-3 intersect circle?

(The description of the graph: Just imagine the circle with its center corresponding to 0 of the XY-plane)  

Answer
Hi Aes,
The equation of the circle is
x^2 + y^2 = 4
and the equation of the parabola is
y = x^2 - 3
Substitutuing y = x^2 - 3 into the first equation to get all the points of intersection gives
x^2 + (x^2 - 3)^2 = 4
x^2 + (x^4 - 6x^2 + 9) = 4
x^4 - 5x^2 + 5 = 0
This is a quadratic expression in x^2 and can be solved by letting x^2 = k. You get two values for k and hence four values for x also giving you four values for y.
So there should be four points of intersection.
I think you can now go ahead and complete the solution. Just let me know if you need anything else. Try drawing the situation on graph and compare your answers. You can also try to go over the whole of the calculations all by yourself.
Good luck.
Regards.