Chapter 15: Problem 1363
If two circle \((\mathrm{x}-1)^{2}+(\mathrm{y}-3)^{2}=\mathrm{a}^{2}\) and \(\mathrm{x}^{2}+\mathrm{y}^{2}-8 \mathrm{x}+2 \mathrm{y}+8=0\) intersect in two distinct points, then (a) \(2<\mathrm{a}<8\) (b) \(a>2\) (c) \(\mathrm{a}<2\) (d) \(\mathrm{a}=2\)
Short Answer
Expert verified
The correct option is (b) \(a > 2\). The two circles intersect at two distinct points when the sum of their radii is more than the distance between their centers, which corresponds to the inequality \(a > 2\).
Step by step solution
01
Identify the center and radius of each circle
The equation of a circle can be written in the form \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius of the circle.
For the first circle, we can see that it has the standard form with \(h = 1\), \(k = 3\), and \(r = a\). So, the center is \((1, 3)\) and the radius is \(a\).
For the second circle, we need to rewrite its equation in the standard form. We can do this by completing the square.
02
Rewrite the second circle's equation in the standard form
The equation given is \(x^2 + y^2 - 8x + 2y + 8 = 0\). To rewrite this equation in the standard form, complete the square for the terms involving \(x\) and \(y\).
1. Add and subtract \((\frac{8}{2})^2 = 16\) inside the equation.
2. Add and subtract \((\frac{2}{2})^2 = 1\) inside the equation.
The equation becomes: \((x^2 - 8x + 16) + (y^2 + 2y + 1) = 9\)
Now, the equation is in the standard form, and we can identify the center and radius of the circle: \(h = 4\), \(k = -1\), and \(r = 3\). So, the center is \((4, -1)\) and the radius is \(3\).
03
Analyze the relative position of circles
Now that we have the centers \((1, 3)\) and \((4, -1)\) and radii \(a\) and \(3\) of both circles, we can check for their intersection points.
Let's find the distance between the centers using the distance formula:
\(d = \sqrt{(4-1)^2 + (-1-3)^2} = \sqrt{9 + 16} = 5\)
Now, there are three possible cases for the relative position of circles:
1. If the sum of their radii is less than the distance between their centers, then the circles do not intersect.
In this case, we have: \(a + 3 < 5\).
2. If the sum of their radii is equal to the distance between their centers, then the circles touch each other externally.
In this case, we have: \(a + 3 = 5\).
3. If the sum of their radii is more than the distance between their centers, then the circles intersect at two distinct points.
In this case, we have: \(a + 3 > 5\).
Since we are interested in when the circles intersect at two distinct points, we will focus on the third case:
\(a + 3 > 5\)
04
Find the range of a for which circles intersect
Now, to find the range of \(a\) where the circles intersect, we need to solve the inequality in Step 3:
\(a + 3 > 5 \Rightarrow a > 2\)
So for \(a > 2\), the circles intersect at two distinct points. Therefore, the correct option is (b) \(a > 2\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Geometry Problem Solving
Geometry problem solving often requires a methodical approach to analyze and solve for the characteristics of geometric shapes, including their size, position, and the relationship between different shapes. In the case of circle intersection problems, it's important to first understand the standard equation of a circle and how it represents a circle's basic properties, such as its center and radius.
In the given exercise, solving for the intersection points between two circles involves determining the center and radius of each circle and analyzing their relative positions. This is fundamental for predicting whether they intersect, touch, or are separate from each other. Students must identify known and unknown variables associated with the circles and apply mathematical concepts like the distance formula to find the exact conditions leading to their intersection. The process of identifying these variables and solving for them step by step ensures a logical and structured route to problem-solving.
In the given exercise, solving for the intersection points between two circles involves determining the center and radius of each circle and analyzing their relative positions. This is fundamental for predicting whether they intersect, touch, or are separate from each other. Students must identify known and unknown variables associated with the circles and apply mathematical concepts like the distance formula to find the exact conditions leading to their intersection. The process of identifying these variables and solving for them step by step ensures a logical and structured route to problem-solving.
Completing the Square
Completing the square is a crucial algebraic technique used to solve quadratic equations, manipulate algebraic expressions, and rewrite the equation of a circle into its standard form. The method involves creating a perfect square trinomial from the original quadratic equation or expression, which simplifies the process of finding its roots or vertex.
In the context of circle equations, completing the square is used to identify the circle's center \(h,k\) and radius \(r\). This is done by rearranging the circle equation to the form \( (x-h)^2 + (y-k)^2 = r^2 \), where the terms inside the parentheses form perfect squares. In the provided exercise, completing the square enabled us to transform the given circle equation into one that visually expresses the circle's geometric properties, which is essential for further analysis and finding the solution to the geometry problem.
In the context of circle equations, completing the square is used to identify the circle's center \(h,k\) and radius \(r\). This is done by rearranging the circle equation to the form \( (x-h)^2 + (y-k)^2 = r^2 \), where the terms inside the parentheses form perfect squares. In the provided exercise, completing the square enabled us to transform the given circle equation into one that visually expresses the circle's geometric properties, which is essential for further analysis and finding the solution to the geometry problem.
Circle Equations
Circle equations are mathematical representations of circles in the coordinate plane. In algebraic terms, the standard form of a circle's equation is \( (x - h)^2 + (y - k)^2 = r^2 \), where \(h\) and \(k\) are the \(x\) and \(y\) coordinates of the circle's center, and \(r\) is the radius. Understanding this equation is essential for solving any problems involving circles, including determining whether and how they intersect.
The given exercise asked us to find the condition for the intersection of two circles. With their equations in standard form, we compared the distance between the centers to the sum of their radii, applying the concept of circle equations to determine how they interact spatially on the coordinate plane. The circle equations thus serve not only as a starting point for identifying the basic properties of the circles but also for exploring the deeper relationship between them in geometric terms.
The given exercise asked us to find the condition for the intersection of two circles. With their equations in standard form, we compared the distance between the centers to the sum of their radii, applying the concept of circle equations to determine how they interact spatially on the coordinate plane. The circle equations thus serve not only as a starting point for identifying the basic properties of the circles but also for exploring the deeper relationship between them in geometric terms.