HPC Ch7 practice

The points [math](0,8)[/math] and [math](-8,4)[/math] lies on a circle. If the center of the circle is the point [math](m,n)[/math], then find [math]\mid 2m+n \mid [/math].

Find all [math]k[/math] so that [math](1,2k), (3k,4)[/math], and [math](5,6k)[/math] cannot determine 3 points on a circle.

Find the equation of the parabola given the following properties: Directrix of [math]x=-3[/math], focus at [math](-1,2)[/math].

The latus rectum of a parabola measures 8, and the parabola’s vertex is the point [math](-4,2)[/math]. If the parabola opens to the right, then how many quadrants does the parabola lie?

Write the center,foci, and vertices of the ellipse [math]9y^2+36y+25x^2-50x-164=0[/math].

Write an equation for the ellipse

Using [math](4,0)[/math] as the focus and the vertical line [math]x=9[/math] as the directrix, find an equation for the set of all points [math](x,y)[/math] where the distance from the focus to [math](x,y)[/math] is two-thirds the distance from the directrix to [math](x,y)[/math].

Consider the hyperbola with equation [math]\dfrac{(y-3)^2}{20}-\dfrac{(x+1)^2}{5}=1[/math]. Let [math]d[/math] be the [math]y[/math]-intercept of the asymptote with a positive slope. Let [math]g[/math] be the sum of the [math]y[/math]-coordinates of the foci. Evaluate [math]\mid d-g \mid[/math].

The equation [math]4x^2-y^2-24x+4y-32=0[/math] represents a conic section. Find the distance between the foci.

If the eccentricity of the ellipse [math]16x^2+9y^2-128x+54y-239=0[/math] is [math]\dfrac{\sqrt{b}}{a}[/math] where [math]a[/math] and [math]b[/math] are integers, what is the value of [math]a+b[/math]?

Let [math]T[/math] be the set of three conics sections. How many element of [math]T[/math] must have an eccentricity of at most 1?

Which of the following values of [math]t[/math] produces the same point as [math]t=\dfrac{2\pi}{3}[/math] in the parametric form [math]x=2\cos t, y=2\sin t[/math]?

Which of the following describes the graph of the parametric equation [math]x=1-t, y=3t+2, t\geq0[/math]?

Find the standard form of the equation of the parabola with vertex [math](2,1)[/math] and focus [math](2,4)[/math].

Find the focus of the parabola [math]y=-\dfrac{1}{2}x^2-x+\dfrac{1}{2}[/math].

Determine the equation of the given conic section from the graph.

Parabola that opens up has a vertex at [math](-2,-4)[/math] and contains point [math](-4,3)[/math]. Determine the equation of the directrix of this parabola.

You installed a parabolic satellite dish on your house to watch satellite TV. The satellite dish is 5.8 inches deep and 16 inches wide. If the receiver is placed at the focus, how far from the vertex of the satellite dish should you place the receiver?

The vertex of [math](y-3)^2=-8(x+2)[/math] is

Find the directrix of the parabola: [math]y^2-x=0[/math]

Find the focus of the parabola: [math]y^2-6y-12x-15=0[/math]

Find an equation of the parabola whose graph is shown.

Write an equation for the ellipse:

Find the standard form of the equation of the ellipse centered at [math](2,-1)[/math] with a vertex [math]\left(2,\dfrac{1}{2}\right)[/math] and a minor axis of length 2.

Find the standard form of the equation of the ellipse with foci at [math](0,0)[/math] and [math](4,0)[/math] and a major axis of length 8.

(Calculator) The opening of a tunnel is in the shape of the upper half of an ellipse 152 feet wide and 38 feet high. Find the height, in feet, of the arch 19 feet from the center of the tunnel.

The perimeter of a triangle with one vertex on the ellipse [math]\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1[/math] and the other two vertices at the foci of the ellipse would be

Find the center of the ellipse: [math]x^2+2y^2-4x-16y+32=0[/math]

Find the vertices of the ellipse: [math]\dfrac{(x+\frac{1}{2})^2}{8}+\dfrac{(y-\frac{1}{2})^2}{4}=1[/math]

Find the standard form of the equation of the ellipse with foci [math](-\sqrt{11},5)[/math] and [math](\sqrt{11},5)[/math] and a vertex [math](6,5)[/math]

(Calculator) A carpenter wishes to construct an elliptical table top form a 4ft by 8ft sheet of plywood. He will trace out the ellipse using the “thumbtack and string” method. What length of string should be use, and how far apart should the tacks be located, if the ellipse is to be the largest possible that can be cut of the plywood sheet?

Write the equations of the asymptotes for the hyperbola [math]\dfrac{y^2}{49}-\dfrac{x^2}{9}=1[/math].

The slope of the asymptotes of the hyperbola [math]\dfrac{x^2}{4}-\dfrac{y^2}{3}=1[/math]

Identify the graph of [math]\dfrac{(x-3)^2}{2}-\dfrac{(y+1)^2}{9}=1[/math]

Find the standard form of the equation for the hyperbola with center [math](0,-2)[/math], one vertex [math](0,-4)[/math], and one focus [math](0,2)[/math].