Examples of distance formula 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 between points $(x_{1},y_{1})$ and $(x_{2},y_{2})$ is given by the formula:
- Substitute the values into the distance formula that is derived from the Pythagorean Theorem:
- The distance formula between two points, $(x_{1},y_{1})$ and $(x_{2},y_{2})$, shown as the hypotenuse of a right triangle
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- The equation for a circle is just an extension of the distance formula.
- To find a formula for this, suppose that the center is the point (a,b).
- According to the distance formula, the distance c from the point (a,b) to any other point (x, y) is
- This is the general formula for a circle with center (a,b) and radius r.
- Notice that all we have done is slightly rearrange the distance formula equation.
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- There are formulae for the addition and subtraction of angles within each of the trigonometric functions.
- To see how these formulae are derived, we can place points on a diagram of a unit circle.
- The angles are equal, and so the distance between points $P$ and $Q$ is the same as between points $A$ and $B$.
- We can derive the following six formulae.
- Apply the formula $\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta$:
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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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- To find out how much it will cost you can use following formula:
- Plugging the known values into the above formula, we can determine that you will pay $500 in interest.
- There are many other common formulas that can be used for everyday computations.
- The formula relating gratuity (G), cost (c), and desired percent gratuity (r, expressed as a decimal).
- Use a given linear formula to solve for a missing variable
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- An ellipse, which resembles an oval, is defined as all points whose distance from two foci add to a 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 general formula for an ellipse is $\frac {x^2}{a^2}+\frac {y^2}{b^2} = 1$.
- Use formulas to determine the area and eccentricity of an ellipse
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- The set of all points in a plane that are the same distance from a given point forms a circle.
- The set of all points in a plane that are the same distance from a given point forms a circle.
- The point is known as the center of the circle, and the distance is known as the radius.
- You already know the formula for a line: y=mx+b.
- To understand the formula below, think of it as the y=mx+b of circles.
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- Complex numbers can be represented in polar coordinates using the formula $a+bi=re^{i\theta}$.
- From this we can see that the product of $z$ and $w$ is the complex numbers whose distance from the origin is the product of the distances from the origin of $z$ and $w$, and whose angle with the horizontal is the sum of the angles of $z$ and $w$ with the origin.
- So $z$ is the complex number which is $\sqrt2$ units from the origin and whose angle with the horizontal is $\pi/4$ radians which is $45 $ degrees, while $w$ is the number whose distance from the origin is $\sqrt2$ and whose angle with the origin is $3\pi/4$ radians which is $135$ degrees.
- Perhaps more easily we could multiply $zw=\sqrt2 e^{i\pi/4}\cdot\sqrt2 e^{3i\pi/4} = 2e^{i\pi} = -2$, realizing that we are getting the number whose distance from the origin is $2$ and whose angle with the horizontal is $\frac{\pi}{4}+\frac{3\pi}{4}=\pi,$ or $180$ degrees.
- Similarly, if $z=re^{i\phi}$ and $w=se^{i\theta}$ then $\frac{z}{w}$ is the result of dividing $\frac{re^{i\phi}}{se^{i\theta}} = \frac{r}{s} e^{i(\phi-\theta)}.$ In other words, when dividing by a complex numbers, the result is a number whose distance from the origin is the quotient of the distances of the two numbers from the origin, and whose angle with the horizontal is the difference of the angles with the horizontal of the two numbers.
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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.
- Because of the minus sign in some of the formulas below, it is also called the imaginary axis of the hyperbola.
- 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).
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