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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

Solution Preview

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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.

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