Problem 2 Find where the circle $$ (x+... [FREE SOLUTION]

Chapter 6: Problem 2

Find where the circle $$ (x+2)^{2}+(y-1)^{2}=9 $$ intersects (a) the $x$ axis, (b) the $y$ axis.

Short Answer

Expert verified

The circle intersects the x-axis at ((-2 + 2鈭�2, 0) and (-2 - 2鈭�2, 0)); the y-axis at (0, 1 + 鈭�5) and (0, 1 - 鈭�5).

Step by step solution

Identify the Circle Equation

The equation of the circle is given by $(x+2)^2 + (y-1)^2 = 9$. This represents a circle with a center at $(-2, 1)$ and a radius of $3$.

Determine Intersection with x-axis

For any point on the $x$-axis, the $y$-coordinate is $0$. Substitute $y = 0$ into the circle's equation to find the $x$-coordinates.\[(x+2)^2 + (0-1)^2 = 9\] Simplify to find \[(x+2)^2 + 1 = 9\] then \[(x+2)^2 = 8\]. Taking the square root gives $x+2 = \pm\sqrt{8}$, so $x = -2 \pm 2\sqrt{2}$. Thus, the points are $(-2 + 2\sqrt{2}, 0)$ and $(-2 - 2\sqrt{2}, 0)$.

Determine Intersection with y-axis

For any point on the $y$-axis, the $x$-coordinate is $0$. Substitute $x = 0$ into the circle's equation to find the $y$-coordinates.\[(0+2)^2 + (y-1)^2 = 9\] Simplify to find \[4 + (y-1)^2 = 9\] then \[(y-1)^2 = 5\]. Taking the square root gives $y-1 = \pm\sqrt{5}$, so $y = 1 \pm \sqrt{5}$. Thus, the points are $(0, 1 + \sqrt{5})$ and $(0, 1 - \sqrt{5})$.

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Key Concepts

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

Equation of a Circle

Understanding the equation of a circle is key in analytic geometry. The standard form of the equation for a circle is \[ (x - h)^2 + (y - k)^2 = r^2 \]where

$(h, k)$ is the center of the circle, and
$r$ is the radius.

In the given exercise, the circle's equation is $(x+2)^2 + (y-1)^2 = 9$.This can be rewritten as $(x - (-2))^2 + (y - 1)^2 = 3^2$, revealing the center as $(-2, 1)$ and the radius as $3$.Recognizing this form helps quickly identify these key components. Every circle has a unique center and radius, making this standard form incredibly useful for graphing and solving problems related to circles.

X-axis Intersection

Finding the intersection of a circle with the x-axis involves setting the y-coordinate to zero, since every point on the x-axis has $y = 0$.To solve for the x-coordinates, substitute $y = 0$ into the circle's equation:\[(x+2)^2 + (0-1)^2 = 9\].This simplifies down to \[(x+2)^2 + 1 = 9\], and further to \[(x+2)^2 = 8\].Upon solving, we find $x+2 = \pm \sqrt{8}$. Therefore, the solutions for $x$ are $-2 + 2\sqrt{2}$ and $-2 - 2\sqrt{2}$.These are the x-coordinates where the circle intersects the x-axis, resulting in the intersection points $(-2 + 2\sqrt{2}, 0)$ and $(-2 - 2\sqrt{2}, 0)$.This principle helps find how shapes interact with axes in coordinate geometry.

Y-axis Intersection

To find where the circle intersects the y-axis, we set the x-coordinate to zero, since all points on the y-axis have $x = 0$.Substituting $x = 0$ into the circle's equation gives us:\[(0+2)^2 + (y-1)^2 = 9\].This simplifies to \[4 + (y-1)^2 = 9\], then further to \[(y-1)^2 = 5\].By solving, we find $y-1 = \pm \sqrt{5}$. Thus, solving for the $y$ value gives $y = 1 + \sqrt{5}$ and $y = 1 - \sqrt{5}$.These are the y-coordinates where the circle intersects the y-axis, leading to the intersection points $(0, 1 + \sqrt{5})$ and $(0, 1 - \sqrt{5})$.This method is a simple way to analyze circle intersections with axes using basic coordinate geometry.

Coordinate Geometry

Coordinate geometry, or analytic geometry, allows us to use algebraic equations to represent geometric figures. In this exercise, the circle's equation and its interactions with coordinate axes provide a perfect example. By representing a circle as an equation, we can algebraically determine how it intersects the x and y axes, translating these algebraic findings into geometric understanding.

The x-axis is represented by setting $y = 0$, making it possible to solve for x-intersections.
Similarly, the y-axis is represented by $x = 0$, helping us find y-intersections.

This powerful approach allows for precise calculations and graphical representations, essential skills in geometry.

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91影视

Find where the circle $$ (x+2)^{2}+(y-1)^{2}=9 $$ intersects (a) the \(x\) axis, (b) the \(y\) axis.

Short Answer

Step by step solution

Identify the Circle Equation

Determine Intersection with x-axis

Determine Intersection with y-axis

Key Concepts

Equation of a Circle

X-axis Intersection

Y-axis Intersection

Coordinate Geometry

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