Conic Sections

Conic Sections: Level 2 Challenges


What is the number of intersections between \(y=4\) and \(x^2+y^2=9\)?

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

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

Point \(O\) is the center of the ellipse with major axis \(AB\) & minor axis \(CD\) Point F is one of the focus of this ellipse.

If \(OF=6\), and the diameter of inscribed circle of triangle \(\triangle{OCF}\) is \(2\), then find \( (AB)\cdot (CD)\)

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

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


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