/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 76 Find an equation of a circle sat... [FREE SOLUTION] | 91影视

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Chapter 13: Problem 76

Find an equation of a circle satisfying the given conditions. Center \((-7,-4)\) and tangent to the \(x\) -axis

Short Answer

Expert verified
(x + 7)^2 + (y + 4)^2 = 16.

Step by step solution

01

Understand the Circle Equation

The general equation of a circle with center \( (h, k) \) and radius \( r \) is given by: \[ (x - h)^2 + (y - k)^2 = r^2 \].
02

Identify the center

Given the center of the circle is \((-7, -4)\), we identify \(h = -7\) and \(k = -4\).
03

Determine the radius of the circle

Since the circle is tangent to the \(x\)-axis, the distance from the center to the \(x\)-axis is the radius. Using the center coordinates \(( -7, -4 )\), the radius \(r\) is the absolute value of the \(y\)-coordinate of the center, which is \(|-4| = 4\).
04

Plug values into the circle equation

Substitute \(h = -7\), \(k = -4\), and \(r = 4\) into the general equation: \[ (x - (-7))^2 + (y - (-4))^2 = 4^2 \].
05

Simplify the equation

Simplify the equation: \[ (x + 7)^2 + (y + 4)^2 = 16 \]. This is the equation of the circle.

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Key Concepts

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

circle equation
A circle can be defined by its equation in geometry. The general equation of a circle with center \( (h,k) \) and a radius \( r \) is given by the formula: \[ (x - h)^2 + (y - k)^2 = r^2 \]. This represents all the points \( (x,y) \) that are at a distance \( r \) from the center. You can think of \( h \) and \( k \) as the coordinates of the center point. Knowing this formula makes it easier to handle various problems involving circles.
radius
The radius of a circle is the distance from the center to any point on the circumference. In the equation of a circle, \( r \) represents this radius. For the given problem, where the circle is tangent to the x-axis, the radius can be easily determined. The center is at \((-7, -4)\). Since the circle touches the x-axis, the radius is simply the absolute value of the y-coordinate of the center, which is \(| -4 | = 4\). This distance tells us how far the circle extends from the center to the point where it's tangent to the x-axis.
center coordinates
The center coordinates of a circle are crucial for writing its equation. For a circle centered at \( (h, k) \), these coordinates are plugged into the general circle equation. In our problem, the center is \((-7, -4)\). Identifying these values correctly is key to solving the equation. Once we know the center, we can substitute \( h \) and \( k \) with \(-7\) and \(-4\) respectively in the equation \[(x - h)^2 + (y - k)^2 = r^2\]. This substitution helps us move forward to finding the complete equation.
tangent to x-axis
When a circle is tangent to the x-axis, it means the circle touches the x-axis at exactly one point. This contact point informs us about the radius. The distance from the center to the touching point on the x-axis is the radius. For our circle with center at \((-7, -4)\), this distance is simply the absolute value of the y-coordinate, which calculates to \(| -4 | = 4\). So, when given that a circle is tangent to an axis, remembering this can simplify finding the radius. It follows naturally that the circle's equation will reflect this distance accurately.

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Most popular questions from this chapter

Solve. \(5 x+2 y=-3\) \(2 x+3 y=12\)

Find the center and the radius of each circle. Then graph the circle. $$ 4 x^{2}+4 y^{2}=1 $$

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Solve $$ \begin{aligned} &a+b=\frac{5}{6}\\\ &\frac{a}{b}+\frac{b}{a}=\frac{13}{6} \end{aligned} $$
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