# Identifying types of conics

For each of the equations identify the type of conic it is and list the major features associated with this type of conic

Major Features

Circles

- Find the coordinates of the center

- Find the length of the radius

- Find the length of the diameter

Parabolas

- Find the coordinates of the vertex

- Find the coordinates of the focus

- Write the equation of the axis of symmetry

- Write the equation of the directrix

Ellipses

-Find the center of the ellipse

- Fine the coordinates of both of the foci points

- Find the coordinates of alll the vertices

- Find the length of the major axis

-Find the length of the minor axis

Hyperbolas

- Find the center

-Find the coordinates of the twp vertices

-Find the coordinates of the two foci points

- Find the equation of the two asymptotes

Equations

1) X62 - 10x +2y^2 +40y - 175=0

2) 2x^2 + 6x + 5y^2 - 40y - 7 = 0

3) 5x^2 - 80x + 5y^2 - 30y - 135 = 0

4) 6x^2 + 24x + 3y - 45 = 0

https://brainmass.com/math/synthetic-geometry/identifying-types-of-conics-15043

#### Solution Preview

2.26

1. x^2 - 10x + 2y^2 + 40y -175 = 0 Rewriting...

(x^2-10x) + 2 (y^2+20y)- 175 = 0

Completing the square...

(x^2-10x+25)-25 + 2(y^2+20y+100)-100 -175 = 0

(x-5)^2 + 2(y+10)^2 = 200 Putting in a standard form...

(x-5)^2/200 + (y+10)^2/100 = 1

This is an ellipse with major axis a = 200 and minor axis b = 100

Center is at (5,-10)

Foci are (+ -a e,0)

We have b^2 = a^2 [1-e^2] from this e = sqrt[1- b^2/a^2] = sqrt(3)/2

Foci are, (+ - 100*sqroot(3),0)

That is, x-5 = +- 100* sqroot(3)==> x = (+ -) 100*sqrt(3)+5

and y+10 = 0 ==> y = -10 are the co- ordinates of ...

#### Solution Summary

This provides four equations and shows how to identify which type of conic section each one is.

Conic Section is explored.

Equation Point

4x^2-24x-25y^2+250y-489=0 (27⁄4,5⁄2)

( 1) identify the conic section represented by the equation.

( 2) write the equation of the conic section in standard form.

( 3) identify all relevant key elements of your conic section such as center,

focus/foci, directrix, radius, lengths of major and/or minor axes,

equations of asymptotes, and length of latus rectum.

( 4) sketch the graph of the conic section.

( 5) determine the equation of the line that is tangent to the conic section

at the provided point.

( 6) write a brief (two to three paragraph) essay describing the four nondegenerate

conic sections, how you knew to distinguish your conic section from the

other three, and a possible application that might involve your type of

conic section