Conic Sections

Conic Sections

Practice identifying characteristics of conic sections from graphs and equations

Multiple Choice

A hyperbola centered at (-3, 2) with a horizontal tranverse axis has asymptotes y = ± ¾x. Which of the following could be the equation of the hyperbola in standard form?

• $$\frac{\left(x+3\right)^2}{16}+\frac{\left(y-2\right)^2}{9}=1$$

• $$\frac{\left(y-2\right)^2}{9}-\frac{\left(x+3\right)^2}{16}-=1$$

• $$\frac{\left(x+3\right)^2}{16}-\frac{\left(y-2\right)^2}{9}=1$$

• $$\frac{\left(x+3\right)^2}{9}-\frac{\left(y-2\right)^2}{16}=1$$

Multiple Choice

Write the standard form of the Hyperbola that has vertices at (0, 13) and (0, -15) and asymptotes at $$y=\frac{7}{5}x-1$$  and  $$y=-\frac{7}{5}x-1$$ . 

• $$\frac{\left(y+1\right)^2}{196}-\frac{x^2}{100}=1$$  

• $$\frac{y^2}{196}-\frac{\left(x-1\right)^2}{100}=1$$  

• $$\frac{x^2}{100}-\frac{\left(y-1\right)^2}{196}=1$$  

• $$\frac{x^2}{196}-\frac{\left(y+1\right)^2}{100}=1$$  

Multiple Choice

Find the equation of hyperbola.

• $$\frac{\left(x-2\right)^2}{16}-\frac{\left(y+3\right)^2}{20}=1$$

• $$\frac{\left(x+3\right)^2}{16}-\frac{\left(y-2\right)^2}{20}=1$$

• $$\frac{\left(x-2\right)^2}{20}-\frac{\left(y+3\right)^2}{16}=1$$

• $$\frac{\left(y+3\right)^2}{16}-\frac{\left(x-2\right)^2}{20}=1$$

Multiple Choice

Find the standard form of the Hyperbola given that $$25x^2-4y^2-250x+16y+209=0$$ . 

• $$\frac{\left(x-5\right)^2}{16}-\left(y-2\right)^2=1$$  

• $$\frac{\left(y-2\right)^2}{100}-\frac{\left(x-5\right)^2}{16}=1$$  

• $$\frac{\left(x-5\right)^2}{16}-\frac{\left(y-2\right)^2}{100}=1$$  

• $$\frac{\left(x-5\right)^2}{100}-\frac{\left(y-2\right)^2}{16}=1$$  

Multiple Choice

Identify the asymptotes of the equation: $$\frac{\left(y-1\right)^2}{64}-\frac{\left(x+9\right)^2}{36}=1$$  

• $$y=\frac{4}{3}x\ +\ 13\ \ ;\ y=-\frac{4}{3}x\ -\ 11$$  

• $$y=\frac{4}{3}x\ +\ \frac{23}{3}\ ;\ y=-\frac{4}{3}x\ +\ \frac{31}{3}$$  

• $$y=\frac{3}{4}x\ +\ \frac{31}{4}\ ;\ y=-\frac{3}{4}x\ -\ \frac{23}{4}$$  

• $$y=\frac{4}{3}x\ -\ \frac{23}{3}\ ;\ \ y=-\frac{4}{3}x\ -\ \frac{31}{3}$$  

Multiple Choice

What are the coordinates of the Foci ?

• $$\left(-16,2\right),\left(10,2\right)$$

• $$\left(-15,2\right),\left(9,2\right)$$

• $$\left(-3,15\right),\left(-3,-11\right)$$

• $$\left(15,-2\right),\left(-9,-2\right)$$

Multiple Choice

What are the foci of the hyperbola with the equation  $$\frac{y^2}{12}-\frac{x^2}{5}=1$$ .

• $$\left(0,\pm\sqrt{17}\right)$$  

• $$\left(\pm\sqrt{17},0\right)$$  

• $$\left(0,\pm\sqrt{7}\right)$$

• $$\left(\pm\sqrt{7},0\right)$$  

Multiple Choice

What are the co-vertices of the ellipse?

• $$\left(-5,3\right)$$

• $$\left(-5,0\right)\ and\ \left(-5,6\right)$$

• $$\left(3,-5\right)$$

• $$\left(-10,3\right)\ and\ \left(0,3\right)$$

Multiple Choice

Which of the following is the graph of  $$\frac{\left(x+3\right)^2}{4}+\frac{\left(y+1\right)^2}{9}=1$$  

Multiple Choice

Find the standard equation of the given ellipse

Multiple Choice

Determine the center and foci of an ellipse $$4x^2+y^2-16x-10y+25=0$$  

• Center (-2,3) and foci =  $$\left(5,\ 3\pm\sqrt{2}\right)$$  

• Center (4,-6) and foci =  $$\left(2,\ 3\pm\sqrt{4}\right)$$  

• Center (2, 5) and foci =  $$\left(2,\ 5\pm\sqrt{12}\right)$$  

• Center (3,-2) and foci=  $$\left(5,\ 2\pm\sqrt{12}\right)$$  

Multiple Choice

The focus is at (2,0) and the vertex is at (-4,0). What is the equation of the parabola?

• $$y^2=24\left(x+4\right)$$

• $$\left(y+4\right)^2=-12x$$

• $$x^2=16\left(y+4\right)$$

• $$\left(x+4\right)^2=-20y$$

Multiple Choice

(x+3)² = 4(y+5)

What is the equation of the directrix?

• $x=4$

  
  

• $x=2$

• $y=-4$

• $y=-5$

Multiple Choice

What are the coordinates for the focus?

$$\left(y-4\right)^2=-8\left(x+1\right)$$  

• $$\left(1,4\right)$$  

• $$\left(1,-6\right)$$

• $$\left(1,-2\right)$$  

• $$\left(-3,4\right)$$

Multiple Choice

What direction does this parabola open?

$$\left(y-4\right)^2=-8\left(x+1\right)$$  

• up

• down

• left

• right

Multiple Choice

All parabolas with a vertical directrix open the left or right.

• true

• false

Multiple Choice

Given the equation of the parabola,  $y^2-16y+8x-15=0$ where would the VERTEX be? 

•  $\left(3,3\right)$  

  
  

•  $\left(24,3\right)$  

•  $\left(-24,3\right)$  

•  $\left(-3,3\right)$  (-3, 3)

Multiple Choice

$$\left(y-9\right)^2=-40\left(x+4\right)$$  

What direction does the parabola open?

• Up

• Down

• Left

• Right

Multiple Choice

Write an equation for the ellipse with each set of characteristics. Then answer the question. Vertices $$\left(-2,-4\right),\left(-2,8\right)$$   Length of minor axis is 10 What is the center of the ellipse?

• $$\left(-2,2\right)$$

• $$\left(-2,4\right)$$

• $$\left(2,-2\right)$$  

• $$\left(-2,1\right)$$

Multiple Choice

$$\left(y+5\right)^2=-12\left(x-2\right)$$  

What is the focus of the  parabola?

• (2, - 5)

• ( -5, 2)

• (1 , -5 )

• (-1, -5 )

Multiple Choice

$$\left(x-4\right)^2=8\left(y+3\right)$$  

What is the directrix of the parabola?

• y = - 3

• x= 5

• y= - 5

• y= 2

Multiple Choice

Which type of conic section is this equation?

• circle

• ellipse

• hyperbola

• parabola

Multiple Choice

Which type of conic section is this equation?

• circle

• ellipse

• hyperbola

• parabola

Multiple Choice

Which type of conic section is this equation?

• circle

• ellipse

• hyperbola

• parabola

Multiple Choice

Which type of conic section is this equation?

• circle

• ellipse

• hyperbola

• parabola

Multiple Choice

Which type of conic section is this equation?

• circle

• ellipse

• hyperbola

• parabola

Fill in the Blanks

A

is the set of all points equidistant from a focus and a line called the directrix.

Fill in the Blanks

An

is the set of all points such that the distances between a point on the curve and its foci have a constant sum.

Fill in the Blanks

The difference of the distances between a point on a curve and its foci is a constant is the definition of a(n)