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Chapter 1: Problem 75

Write the standard form of the equation of the circle with the given characteristics. Endpoints of a diameter: \((0,0),(6,8)\)

Short Answer

Expert verified
The standard form of the equation for the circle with endpoints of a diameter at (0,0) and (6,8) is (x-3)^2 + (y-4)^2 = 25.

Step by step solution

01

Find the midpoint

The midpoint of the diameter can be calculated using the midpoint formula (d1 + d2) / 2. Let's denote the input endpoints as (x1,y1) and (x2,y2). So the midpoint, which is the center of the circle, has coordinates: cx = (x1 + x2) / 2 = (0 + 6) / 2 = 3 and cy = (y1 + y2) / 2 = (0 + 8) / 2 = 4
02

Calculate the Radius

The radius of the circle is the distance from the center to either endpoint. To get the radius, apply the distance formula r = sqrt[(x2-x1)^2 + (y2-y1)^2] / 2 . Now we'll plug in the center and one of the endpoints. Here, radius r = sqrt[(6-3)^2 + (8-4)^2] = sqrt[9 + 16] = sqrt[25] = 5
03

Write the Standard Form of the Circle's Equation

The standard form of a circle's equation is (x-h)^2 + (y-k)^2 = r^2, where (h,k) is the center and r is the radius. Substituting the values for the center (3,4) and radius 5, the standard form equation for our circle becomes (x-3)^2 + (y-4)^2 = 5^2

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Most popular questions from this chapter

Intercept Form of the Equation of a Line In Exercises \(81-86,\) use the intercept form to find the equation of the line with the given intercepts. The intercept form of the equation of a line with intercepts \((a, 0)\) and \((0, b)\) is \(\frac{x}{a}+\frac{y}{b}=1, a \neq 0, b \neq 0\) Point on line: \((1,2)\) \(x\) -intercept: \((c, 0)\) \(y\) -intercept: \((0, c), \quad c \neq 0\)

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The table shows the life expectancies of a child (at birth) in the United States for selected years from 1930 through \(2000\) . $$\begin{array}{|c|c|}\hline \text { Year } & \text { Life Expectancy, \(y\) } \\\\\hline 1930 & 59.7 \\\1940 & 62.9 \\\1950 & 68.2 \\\1960 & 69.7 \\\1970 & 70.8 \\\1980 & 73.7 \\\1990 & 75.4 \\\2000 & 76.8 \\\\\hline\end{array}$$ A model for the life expectancy during this period is $$y=-0.002 t^{2}+0.50 t+46.6, \quad 30 \leq t \leq 100$$ where \(y\) represents the life expectancy and \(t\) is the time in years, with \(t=30\) corresponding to 1930. (a) Use a graphing utility to graph the data from the table and the model in the same viewing window. How well does the model fit the data? Explain. (b) Determine the life expectancy in 1990 both graphically and algebraically. (c) Use the graph to determine the year when life expectancy was approximately \(76.0 .\) Verify your answer algebraically. (d) One projection for the life expectancy of a child born in 2015 is \(78.9 .\) How does this compare with the projection given by the model? (e) Do you think this model can be used to predict the life expectancy of a child 50 years from now? Explain.
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