Conic Sections

Conic Sections: Level 2 Challenges


What is the number of intersections between y=4y=4 and x2+y2=9x^2+y^2=9?

For any real number α \alpha, the parabola fα(x)=2x2+αx+3α f_\alpha (x) = 2x^2 + \alpha x + 3 \alpha passes through the common point (a,b) (a, b) . What is the value of a+b?a+b?

Consider the point P(1,0)P(-1, 0) on the ellipse given by the equation 4x2+y2=44x^2 + y^2 = 4. There are two points (a,b)(a, b) and (a,c)(a, c) on the ellipse whose distance from PP is a maximum. What is the value of aa?

Point OO is the center of the ellipse with major axis ABAB & minor axis CDCD Point F is one of the focus of this ellipse.

If OF=6OF=6, and the diameter of inscribed circle of triangle OCF\triangle{OCF} is 22, then find (AB)(CD) (AB)\cdot (CD)

There are four lines that are tangent to both circles
x2+y2=1 and (x6)2+y2=4. { x }^{ 2 }+{ y }^{ 2 }=1 \quad \text{ and } \quad ({ x }-6)^{ 2 }+{ y }^{ 2 }=4.

What is the sum of the slopes of these four lines?


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