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Chapter 9: Problem 86

Find the equation of the circle passing through the given points. $$(-1,5),(6,6), \text { and }(7,-1)$$

Short Answer

Expert verified
The equation of the circle passing through the points is given by the equation .

Step by step solution

01

Standard form of the equation of a circle

The standard form of the equation of a circle is
02

Write the general form of the circle

The formula for the equation of a circle is The coordinates of the center are given as and The equation of the circle can be given in terms of and

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Form of a Circle's Equation
The standard form of the equation of a circle might appear complicated at first glance, but it鈥檚 straightforward once you understand its components. The standard form is \( (x - h)^2 + (y - k)^2 = r^2 \). This equation represents a circle centered at \( (h, k) \) with radius \( r \).
Unpacking this form, \( (x - h) \) and \( (y - k) \) signify the horizontal and vertical distances from the center to any point \((x, y)\) on the circle. Here鈥檚 a simpler breakdown:
  • \( (h, k) \) is the center of the circle.
  • \ r \ is the radius, the distance from the center to any point on the circle.
Understanding this makes it easier to visualize and write equations for various circles.
Finding the Circle Through Given Points
To find the equation of a circle passing through specific points, such as \((-1,5), (6, 6), \text{and} (7,-1)\), we use coordinate geometry techniques. First, identify that a circle through these points means all points satisfy the circle's equation.
We use the general form \(x^2 + y^2 + Dx + Ey + F = 0 \), then plug in the given points to find \(D \), \(E \), and \(F \). This system of equations can be solved using substitution or elimination methods.
After solving for \ D \, \ E \, and \ F \, we can rewrite the obtained values back into the general equation. This algebraic process might be lengthy, but each step gets you closer to the circle's equation.
Exploring Coordinate Geometry
Coordinate geometry, or analytic geometry, blends algebra and geometry to study geometric figures using a coordinate system. For circles, this system helps seamlessly transition between geometric shapes and algebraic equations.
In coordinate geometry:
  • Points are represented as coordinates \( (x, y) \) on a plane.
  • A circle's properties become equations that algebra can solve.
Studying circles through this lens allows for precise calculations and deeper understanding of geometric properties. By using coordinate geometry, solving real-world problems involving circles, lines, and other shapes becomes much more approachable and accurate.

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Most popular questions from this chapter

Solve each problem. Yogurt sells three types of yogurt: nonfat, regular, and super creamy, at three locations. Location I sells 50 gal of nonfat, 100 gal of regular, and 30 gal of super creamy each day. Location II sells 10 gal of nonfat, and Location III sells 60 gal of nonfat each day. Daily sales of regular yogurt are 90 gal at Location II and 120 gal at Location III. At Location II, 50 gal of super creamy are sold each day, and 40 gal of super creamy are sold each day at Location III. (a) Write a \(3 \times 3\) matrix that shows the sales figures for the three locations, with the rows representing the three locations. (b) The incomes per gallon for nonfat, regular, and super creamy are \(\$ 12, \$ 10,\) and \(\$ 15,\) respectively. Write a \(1 \times 3\) or \(3 \times 1\) matrix displaying the incomes. (c) Find a matrix product that gives the daily income at each of the three locations. (d) What is Yagel's Yogurt's total daily income from the three locations?

Use the shading capabilities of your graphing calculator to graph each inequality or system of inequalities. $$y \leq x^{2}+5$$

Solve each system by using the inverse of the coefficient matrix. $$\begin{aligned} &x+3 y=-12\\\ &2 x-y=11 \end{aligned}$$

Find the cofactor of each element in the second row for each determinant. $$\left|\begin{array}{rrr} -2 & 0 & 1 \\ 1 & 2 & 0 \\ 4 & 2 & 1 \end{array}\right|$$

Supply and Demand In many applications of economics, as the price of an item goes up, demand for the item goes down and supply of the item goes up. The price where supply and demand are equal is the equilibrium price, and the resulting sup. ply or demand is the equilibrium supply or equilibrium demand. Suppose the supply of a product is related to its price by the equation $$p=\frac{2}{3} q$$ where \(p\) is in dollars and \(q\) is supply in appropriate units. (Here, \(q\) stands for quantity.) Furthermore, suppose demand and price for the same product are related by $$p=-\frac{1}{3} q+18$$ where \(p\) is price and \(q\) is demand. The system formed by these two equations has solution \((18,12),\) as seen in the graph. (GRAPH CANNOT COPY) Find the demand for the electric can opener at each price. (a) \(\$ 6\) (b) \(\$ 11\) (c) \(\$ 16\)
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