Conic Sections and Parametric Equations: math test, Grade 12 – download the PDF

1 Circle / 4 pts

Consider \(x^2+y^2+6x-10y+18=0\).

1. Write it in standard form and give the center and radius.
2. Is the point \((-3,9)\) on the circle?
3. Find where the circle meets the line \(y=5\).

2 Parabola / 4 pts

Consider \(y^2=16x\).

1. Give the vertex, focus and directrix.
2. Find the points with \(x=9\) and show that each is 13 units from the focus.

3 Ellipse / 4 pts

An ellipse has equation \(16x^2+25y^2=400\).

1. Write it in standard form and find \(a\), \(b\), \(c\) and the vertices.
2. Give the foci and the eccentricity.
3. The point \(P\left(3,\dfrac{16}{5}\right)\) lies on the ellipse. Compute the sum of its distances to the foci.

4 Hyperbola / 4 pts

Consider \(\dfrac{y^2}{36}-\dfrac{x^2}{64}=1\).

1. Find \(a\), \(b\), \(c\) and the eccentricity.
2. Give the vertices, the foci and the asymptotes.

5 Parametric equations / 4 pts

1. Eliminate \(t\) from \(x=1+4\cos t,\ y=-2+3\sin t\), name the curve and give its center, and the point for \(t=\dfrac{\pi}{2}\).
2. Eliminate \(t\) from \(x=3t-1,\ y=t^2\).