Question 1

Given the equation $$7x^2$$ + $$7y^2 = 112$, identify the type of conic section, write its standard form, and determine its center and radius.

Accepted Answer

It is a circle; standard form: $$x^2$$ + $$y^2 = 16$; center at (0$$,0)$$, radius 4$

Question 2

Given the equation $$16x^2$$ + $$81y^2 + 160x + 216y + 400 = 0$, what type of conic section is it, and what is its center?

Accepted Answer

Ellipse; center at $$(-5, -1.33)$$

Question 3

A circle has an area of $$48\pi$$ and its center is at $$(-3,7). $$What is its equation in standard form?

Accepted Answer

Question 4

Given the endpoints of a circle's diameter at $$(-9,1)$$ and $$(1,5)$$, what is the equation of the circle in standard form?

Accepted Answer

$$(x+4)^2$$ + $$(y-3)^2 = 25$

Question 5

An ellipse has vertices at $$(-2,19)$$ and $$(-2,-7)$$, and co-vertices at $$(7,6)$$ and $$(-11,6). $$What is its standard form equation?

Accepted Answer

$$\frac{(x+2)^2$}{81} + \frac{(y-6)^2$}{169} = 1$$

Question 6

A hyperbola has vertices at $$(15, -10)$$ and $$(-1, -10)$$, and co-vertices at $$(7, -2)$$ and $$(7, -18). $$What is its standard form equation?

Accepted Answer

Question 7

A parabola has vertex $$(4, -6)$$ and focus $$(4, -8). $$What is its equation in standard form?

Accepted Answer

$$(x-4)^2 = -8(y+6$$)$$

Question 8

A planet's semimajor axis is $$3388 Gm $$and its eccentricity $$e = 0.0329. $$What is its aphelion and perihelion?

Accepted Answer

Aphelion: $$3388(1+0.0329) Gm$$, Perihelion: $$3388(1-0.0329) Gm$$

Question 9

A satellite orbits Planet A with the equation $$\frac{x^2$}{(5043)^2$} + \frac{y^2$}{(4775)^2$} = 1$$ and Planet A's center at $$(13,0). If $$Planet A's radius is $$3936$$ miles, what is the perigee height above the surface?

Accepted Answer

Perigee height: $$5043 - 3936 = 1107$$ miles

Question 10

The reflector of a flashlight is a paraboloid with diameter $$4$$ inches and depth $$3$$ inches. How far from the vertex should the light bulb be placed?

Accepted Answer

At $$1$$ inch from the vertex

Question 11

Will a truck that is $$8$$ feet wide and $$7$$ feet tall clear a semielliptical arch that is $$10$$ feet high and $$30$$ feet wide?

Accepted Answer

Yes, because the height at $$4$$ feet from the center is greater than $$7$$ feet

Question 12

A particle moves along a hyperbolic path with an asymptote $$y = \frac{1}{3}x$$ and gets as close as $$7$$ units to the nucleus at $$(0,0). $$What is the equation of its path?

Accepted Answer

$$\frac{x^2$}{49} - \frac{y^2$}{49/9} = 1$$

Question 13

Given the equation $$y^2 - 14x - 8y + 2 = 0$, what type of conic section is it and what is its vertex?

Accepted Answer

Parabola; vertex at $$(7, 4)$$

Question 14

Given the equation $$25x^2$$ - $$9y^2 - 36y - 261 = 0$, identify the conic section and its center.

Accepted Answer

Hyperbola; center at $$(0, 2)$$

Question 15

Given the equation $$x^2$$ + $$y^2 + 8x - 6y + 9 = 0$, what is the center and radius of the circle?

Accepted Answer

Center $$(-4, 3)$$, radius $$2$$

Question 16

A hyperbola has foci at $$(-8, 5)$$ and $$(4, 5)$$, and its domain is $$(-\infty, -5] \cup [1, \infty). $$What is its equation?

Accepted Answer

$$\frac{(x-(-2))^2$}{36} - \frac{(y-5)^2$}{27} = 1$$

Question 17

A parabola is given by $$y = \frac{1}{4}(x-5)^2$ - 2. $$Write the equation for a horizontal parabola with the same vertex and whose focus is twice as far from the vertex.

Accepted Answer

$$(x-5)^2 = 8(y+2$$)$$

Conic Sections Review and Applications – Step-by-Step Precalculus Guidance

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