Question 522542: How can you tell the difference between ellipse, parabola, hyperbola from just looking at the formula? Please explain.
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! How can you tell the difference between ellipse, parabola, hyperbola from just looking at the formula? Please explain.
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To tell the difference between between ellipse, parabola, hyperbola, and to add a circle, the best way is to write the equation for each in its standard form. In most cases, if not in standard form, you must complete the squares to show its standard form. The following are the standard forms for each function:
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Circle: (x-h)^2+(y-k)^2=r^2, (h,k) being the (x,y) coordinates of the center, r=radius.
Ellipse: Two standard forms:
For ellipse with horizontal major axis: (x-h)^2/a^2+(y-k)^2/b^2=1, a>b, (h,k) being the (x,y) coordinates of the center.
For ellipse with vertical major axis: (x-h)^2/b^2+(y-k)^2/a^2=1, a>b, (h,k) being the (x,y) coordinates of the center.
Note that the only difference between the two forms is the interchange of a^2 and b^2, that is, the larger denominator is associated with the major axis, horizontal or vertical.
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Hyperbola:Two standard forms:
For hyperbola with horizontal transverse axis: (hyperbola opens left-right)
(x-h)^2/a^2-(y-k)^2/b^2=1, (h,k) being the (x,y) coordinates of the center.
For hyperbola with vertical transverse axis: (hyperbola opens up-down)
(y-k)^2/a^2-(x-h)^2/b^2=1, (h,k) being the (x,y) coordinates of the center.
Note that the only difference between the two forms is the interchange of (x-h)^2 and (y-k)^2
Also note the form is identical to that of the ellipse except for the negative sign.
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Parabola: several forms:
y=A(x-h)^2+k, (h,k) being the (x,y) coordinates of the vertex, A is just a multiplier which affects the slope or steepness of the curve. Parabola opens upwards, that is, it has a minimum.
y=-A(x-h)^2+k, same as previous form except parabola opens downwards, that is, it has a maximum.
(x-h)^2=4p(y-k), parabola opens upwards showing directrix and focus
(x-h)^2=-4p(y-k), parabola opens downwards showing directrix and focus
(y-k)^2=4p(x-h), parabola opens rightward showing directrix and focus
(y-k)^2=-4p(x-h), parabola opens leftward showing directrix and focus
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