Distance and Midpoint Formulas The distance d between the points (x 1, y 1 ) and (x 2, y 2 ) is...
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Transcript of Distance and Midpoint Formulas The distance d between the points (x 1, y 1 ) and (x 2, y 2 ) is...
![Page 1: Distance and Midpoint Formulas The distance d between the points (x 1, y 1 ) and (x 2, y 2 ) is given by the distance formula: The coordinates of the point.](https://reader036.fdocuments.us/reader036/viewer/2022082610/56649d945503460f94a7cbcc/html5/thumbnails/1.jpg)
Distance and Midpoint Formulas
The distance d between the points (x1, y1) and (x2, y2) is given by the distance formula:
2122
12 yyxxd
The coordinates of the point halfway between the points (x1, y1) and (x2, y2) is given by the midpoint formula:
222121 yy
,xx
• (x1, y1)
• (x2, y2)
•
![Page 2: Distance and Midpoint Formulas The distance d between the points (x 1, y 1 ) and (x 2, y 2 ) is given by the distance formula: The coordinates of the point.](https://reader036.fdocuments.us/reader036/viewer/2022082610/56649d945503460f94a7cbcc/html5/thumbnails/2.jpg)
Conics
Parabolas
Equations of parabolas are quadratic in x or y
2
2
ayx
axy
Vertex at (0, 0)
2
2
kyahx
hxaky
Vertex at (h, k)
![Page 3: Distance and Midpoint Formulas The distance d between the points (x 1, y 1 ) and (x 2, y 2 ) is given by the distance formula: The coordinates of the point.](https://reader036.fdocuments.us/reader036/viewer/2022082610/56649d945503460f94a7cbcc/html5/thumbnails/3.jpg)
Conics
Parabolas
If the x term is quadratic, (ax2), the parabola is vertical.
If a > 0, it opens up
If a < 0, it opens down
![Page 4: Distance and Midpoint Formulas The distance d between the points (x 1, y 1 ) and (x 2, y 2 ) is given by the distance formula: The coordinates of the point.](https://reader036.fdocuments.us/reader036/viewer/2022082610/56649d945503460f94a7cbcc/html5/thumbnails/4.jpg)
Conics
Parabolas
If the y term is quadratic, (ay2), the parabola is horizontal.
If a > 0, it opens right
If a < 0, it opens left
![Page 5: Distance and Midpoint Formulas The distance d between the points (x 1, y 1 ) and (x 2, y 2 ) is given by the distance formula: The coordinates of the point.](https://reader036.fdocuments.us/reader036/viewer/2022082610/56649d945503460f94a7cbcc/html5/thumbnails/5.jpg)
Conics
Circles and Ellipses
Standard form for an ellipse centered at origin:
12
2
2
2
b
y
a
x
-a a
-b
b
![Page 6: Distance and Midpoint Formulas The distance d between the points (x 1, y 1 ) and (x 2, y 2 ) is given by the distance formula: The coordinates of the point.](https://reader036.fdocuments.us/reader036/viewer/2022082610/56649d945503460f94a7cbcc/html5/thumbnails/6.jpg)
Conics
Circles and Ellipses
Standard form for a circle centered at origin with radius r:
222 ryx r-r
r
-r
![Page 7: Distance and Midpoint Formulas The distance d between the points (x 1, y 1 ) and (x 2, y 2 ) is given by the distance formula: The coordinates of the point.](https://reader036.fdocuments.us/reader036/viewer/2022082610/56649d945503460f94a7cbcc/html5/thumbnails/7.jpg)
ConicsHyperbolas
Standard form for hyperbolas centered at origin:
12
2
2
2
b
y
a
x
-a a
12
2
2
2
a
x
b
yb
-b
![Page 8: Distance and Midpoint Formulas The distance d between the points (x 1, y 1 ) and (x 2, y 2 ) is given by the distance formula: The coordinates of the point.](https://reader036.fdocuments.us/reader036/viewer/2022082610/56649d945503460f94a7cbcc/html5/thumbnails/8.jpg)
Conics
If the curve is centered at (h,k), replace the x in the equation with x-h and replace the y with y-k.
222 rkyhx
12
2
2
2
b
ky
a
hx
12
2
2
2
a
hx
b
ky 12
2
2
2
b
ky
a
hx
![Page 9: Distance and Midpoint Formulas The distance d between the points (x 1, y 1 ) and (x 2, y 2 ) is given by the distance formula: The coordinates of the point.](https://reader036.fdocuments.us/reader036/viewer/2022082610/56649d945503460f94a7cbcc/html5/thumbnails/9.jpg)