Conics Part 2

Conics Part 2

Parabolas and maybe Hyperbolas

Multiple Choice

What is the equation of the HORIZONTAL ellipse with a center at (-1, 2), a minor radius of 2 and a major radius of 5.

• $\frac{\left(x+1\right)^2}{25}+\frac{\left(y-2\right)^2}{4}=1$

• $\frac{\left(x+1\right)^2}{4}+\frac{\left(y-2\right)^2}{25}=1$

• $\frac{\left(x-1\right)^2}{25}+\frac{\left(y+2\right)^2}{4}=1$

• $\frac{\left(x-1\right)^2}{4}+\frac{\left(y+2\right)^2}{25}=1$

Multiple Choice

Which is the equation of the ellipse with minor axis endpoints (0, 4) and (0, -4) and foci at (3, 0) and (-3, 0)?

•  $\frac{x^2}{25}+\frac{y^2}{9}\ =\ 1$  

•  $\frac{x^2}{25}+\frac{y^2}{16}\ =\ 1$  

•  $\frac{x^2}{9}+\frac{y^2}{25}\ =\ 1$  

•  $\frac{x^2}{16}+\frac{y^2}{25}\ =\ 1$  

Multiple Choice

For the ellipse located at

 $\frac{\left(x\right)^2}{25}+\frac{\left(y\right)^2}{4}=1$  where are it's foci?

•  $\left(\pm\sqrt{29},0\right)$  

•  $\left(0,\ \pm\sqrt{29}\right)$  

•  $\left(\pm\sqrt{21},0\right)$  

•  $\left(0,\ \pm\sqrt{21}\right)$  

Parabola

• Same as always, BUT parabolas don't JUST open up and down. They can also open left and right

• y = a(x - h)² + k open up and down

• x = a(y - k)² + h open left and right

• a = ¹/_(4p) (I'll get to this more later)

• Vertex is ALWAYS (h, k)

Multiple Choice

Knowing y = a(x - h)² + k , what is the vertex of

y = 4(x + 3)² + 2? and which way does it open?

• (3, 2) up

• (3, 2) down

• (-3, 2) up

• (-3, 2) down

Multiple Choice

Knowing x = a(y - k)² + h , what is the vertex of

x = 4(y + 3)² + 2? and which way does it open?

• (-3, 2) right

• (-3, 2) left

• (2, -3) right

• (2, -3) left

x = a(y - k)² + h

• When a is positive it opens right

• When a is negative it opens left

• In the parenthesis is the y, so the y coordinate of the vertex is in the parenthesis

Now about that ¹/_(4p)

All parabolas are actually....

• The set of points equidistant between a given point (the focus) and a given line (a directrix)

• The vertex is exactly half way between the focus and the directrix

• "p" is the shortest distance between the vertex and either the focus or the directrix (those distances are the same!)

Write the equation of the parabola with directrix x = -2 and focus (4, -1)

• Make a sketch (see the right)

• The focus is ALWAYS inside the parabola and it NEVER crosses the directrix

• Which way will the parabola go?

Multiple Choice

Which way will the parabola go if its directrix is x = -2 and focus (4, -1)?

• Up

• Down

• Left

• Right

Multiple Choice

Which equation will you fill in?

• x = a(y - k)² + h

• y = a(x - h)² + k

Multiple Choice

What will be the vertex of the parabola if its directrix is x = -2 and focus (4, -1)?

• (1, 0)

• (1, -1)

• (-1.5, 4)

• (-1.5, 0)

Open Ended

What does "p" stand for again?

Multiple Choice

If a parabola has a directrix x = -2 and focus (4, -1) and vertex (1, -1), what will "p" be?

• 3

• (3, -1)

• 6

• (6, -1)

Multiple Choice

If p = 3, what is the value of "a"?

• 1/3

• 3/4

• 1/12

• -1/12

If the parabola goes...

Left : "a" is negative

Right: "a" is positive

Down : "a" is negative

Up: "a" is positive

Multiple Choice

If a parabola has a directrix x = -2 and focus (4, -1) and vertex (1, -1), what will the equation be?

• $x\ =\ \frac{1}{12}\left(y\ +\ 1\right)^2\ +\ 1$

• $x\ =\ \frac{1}{12}\left(y\ -\ 1\right)^2\ +\ 1$

• $x\ =\ 3\left(y\ -\ 1\right)^2\ +\ 1$

• $x\ =\ 3\left(y\ +\ 1\right)^2\ +\ 1$

Multiple Choice

If a parabola has a directrix of y = 3 and a focus of (-1, -5), what is the equation of the parabola?

• x = -¹/₈(y - 1)² - 5

• y = -¹/₈(x + 1)² - 1

• x = ¹/₈(y + 5)² + 1

• y = ¹/₈(x + 1)² - 1

Poll

Will you watch the videos and/or practice on Khan to get better at this?

• Yes

• No

Open Ended

Let me know if you need anything!