Examples of distance in the following topics:
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- The distance and the midpoint formulas give us the tools to find important information about two points.
- In analytic geometry, the distance between two points of the $xy$-plane can be found using the distance formula.
- The distance can be from two points on a line or from two points on a line segment.
- The distance between points $(x_{1},y_{1})$ and $(x_{2},y_{2})$ is given by the formula:
- Calculate the midpoint of a line segment and the distance between two points on a plane
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- Absolute value can be thought of as the distance of a real number from zero.
- It refers to the distance of $a$ from zero.
- For example, the absolute value of 5 is 5, and the absolute value of −5 is also 5, because both numbers are the same distance from 0.
- When applied to the difference between real numbers, the absolute value represents the distance between the numbers on a number line.
- Absolute value is closely related to the mathematical and physical concepts of magnitude, distance, and norm.
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- The equation for a circle is just an extension of the distance formula.
- This definition is what gives us the concept of the radius of a circle, which is exactly that certain distance.
- According to the distance formula, the distance c from the point (a,b) to any other point (x, y) is
- Remember that the distance between the center (a,b) and any point (x,y) on the circle is that fixed distance, which is called the radius.
- Notice that all we have done is slightly rearrange the distance formula equation.
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- The orange lines denote the distance between the focus and points on the conic section, as well as the distance between the same points and the directrix.
- These are the distances used to find the eccentricity.
- In other words, the distance between a point on a conic section and its focus is less than the distance between that point and the nearest directrix.
- This indicates that the distance between a point on a conic section the nearest directrix is less than the distance between that point and the focus.
- Examples of these distances are shown.
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- The distance of a directrix from a point on the conic section has a constant ratio to the distance from that point to the focus.
- A conic section is the locus of points $P$ whose distance to the focus is a constant multiple of the distance from $P$ to the directrix of the conic.
- A parabola is the set of all points whose distance from a fixed point, called the focus, is equal to the distance from a fixed line, called the directrix.
- The difference of the distances from any point on the ellipse to the foci
is constant.
- The sum of the distances from any point on the ellipse to the foci is constant.
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- We want the set of all points that have the same difference between the distances to these points.
- Then the difference of distances between $P$ and the two focal points is:
- where $a$ is the distance from the center (origin) to the vertices of the hyperbola.
- With this value for the difference of distances, we can choose any point $(x,y)$ on the hyperbola and construct an equation by use of the distance formula.
- The ellipse can be defined as all points that have a constant sum of distances to two focal points, and the hyperbola is defined as all points that have constant difference of distances to two focal points.
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- An ellipse, which resembles an oval, is defined as all points whose distance from two foci add to a constant.
- An ellipse has the property that, at any point on its perimeter, the distance from two fixed points (the foci) add to the same constant.
- The pen will touch every point on the cardboard such that the distance to one thumbtack, plus the distance to the other thumbtack, is exactly one string length.
- The cardboard is the "plane" in our definition, the thumbtacks are the "foci," and the string length is the "constant distance."
- The sum of the distances from any point P on the ellipse to those two foci is constant and equal to the major axis ( PF1 + PF2 = 2a ).
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- The Cartesian coordinate system is used to specify points on a graph by showing their absolute distances from two axes.
- A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to the two axes.
- Each point can be represented by an ordered pair $(x,y)
$, where the x-coordinate is the point's distance from the y-axis, and the y-coordinate is the distance from the x-axis.
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- At large distances from the center, the hyperbola approaches two lines, its asymptotes, which intersect at the hyperbola's center.
- A hyperbola approaches its asymptotes arbitrarily closely as the distance from its center increases, but it never intersects them.
- The distance b (not shown in below) is the length of the perpendicular segment from either vertex to the asymptotes.
- The eccentricity e equals the ratio of the distances from a point P on the hyperbola to one focus and its corresponding directrix line (shown in green).