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Magazine Chapter 1: Problem 21
In Problems 17-22, find the center and radius of the circle with the given equation. 4 x^{2}+16 x+15+4 y^{2}+6 y=0
Short Answer
Expert verified
Center: (-2, -3/4), Radius: \(\frac{\sqrt{5}}{4}\)
Step by step solution
01
Group Terms
The given equation is \(4x^2 + 16x + 15 + 4y^2 + 6y = 0\). First, we group the terms involving \(x\) and \(y\) separately: \( (4x^2 + 16x) + (4y^2 + 6y) + 15 = 0\).
02
Factor Out Coefficients of Quadratic Terms
Factor out the coefficient of the quadratic terms in each group. For \(x\) terms, factor out 4: \(4(x^2 + 4x)\). For \(y\) terms, factor out 4: \(4(y^2 + \frac{3}{2}y)\).
03
Complete the Square for x
To complete the square for \(x^2 + 4x\), add and subtract \((\frac{4}{2})^2 = 4\) inside the parentheses: \(4((x^2 + 4x + 4) - 4) = 4((x + 2)^2 - 4)\).
04
Complete the Square for y
To complete the square for \(y^2 + \frac{3}{2}y\), add and subtract \((\frac{3}{4})^2 = \frac{9}{16}\) inside the parentheses: \(4((y^2 + \frac{3}{2}y + \frac{9}{16}) - \frac{9}{16}) = 4((y + \frac{3}{4})^2 - \frac{9}{16})\).
05
Simplify Equation
Substitute complete squares and simplify:\[4((x + 2)^2 - 4) + 4((y + \frac{3}{4})^2 - \frac{9}{16}) + 15 = 0\]. Distribute the 4 and combine all constants:\[4(x + 2)^2 - 16 + 4(y + \frac{3}{4})^2 - \frac{9}{4} + 15 = 0\].
06
solve for final equation
Combine all like terms:\[4(x + 2)^2 + 4(y + \frac{3}{4})^2 - 16 - \frac{9}{4} + 15 = 0\] becomes:\[4(x + 2)^2 + 4(y + \frac{3}{4})^2 = 1.25\]. Then divide by 4:\[(x + 2)^2 + (y + \frac{3}{4})^2 = \frac{1.25}{4}\].
07
find center and radius
The equation now has the form \((x - h)^2 + (y - k)^2 = r^2\), where the center \((h, k)\) is \((-2, -\frac{3}{4})\) and radius \(r\) is \(\sqrt{\frac{1.25}{4}} = \frac{\sqrt{1.25}}{2}\). Simplify to find the radius: \(\frac{\sqrt{5}}{4}\)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Completing the Square
Completing the square is a method used to transform a quadratic equation into a perfect square trinomial, making equations easier to work with, especially in a circle's equation.
It involves adding and subtracting a certain number in order to form a complete square.
This technique is essential for graphing and converting equations into the circle's standard form:
This restructuring helps in identifying the center and radius of the circle by making it easier to compare with the equation's standard form.
It involves adding and subtracting a certain number in order to form a complete square.
This technique is essential for graphing and converting equations into the circle's standard form:
- For any quadratic term like \(x^2 + bx\), add and subtract \((\frac{b}{2})^2\) inside the equation.
- Factor any leading coefficient first to make the process simpler.
- Use this transformation to convert quadratic expressions into forms like \((x-h)^2\).
This restructuring helps in identifying the center and radius of the circle by making it easier to compare with the equation's standard form.
Center of a Circle
The center of a circle in its equation's standard form \[(x-h)^2 + (y-k)^2 = r^2\] is identified as the point \((h, k)\).Finding the center is pivotal for understanding the circle's position in a coordinate plane.
Here's how we determine it:
Here's how we determine it:
- After completing the square for both \(x\) and \(y\) sets, the transformation will position them as \((x-h)^2\) and \((y-k)^2\).
- The center coordinates \((h, k)\) are simply the values opposite the constants added inside the completed squares.
For instance, in the equation resulting from our problem, \((x+2)^2 + (y+\frac{3}{4})^2\), the center becomes \((-2, -\frac{3}{4})\).
Radius of a Circle
The radius of a circle is a fundamental part of its geometry.
In the equation \[(x-h)^2 + (y-k)^2 = r^2\], \(r^2\) represents the radius squared.
The calculation of the radius involves:
Remember, the radius extends from the center to any point on the circle, representing the circle's constant distance from the center.
In the equation \[(x-h)^2 + (y-k)^2 = r^2\], \(r^2\) represents the radius squared.
The calculation of the radius involves:
- Converting the constant on the right side of the equation by factoring out and simplifying as necessary.
- Taking the square root of the resulting number to determine \(r\).
- For the example equation, after simplifying, we reached the form \( (x + 2)^2 + (y + \frac{3}{4})^2 = \frac{5}{16} \), with \(r^2 = \frac{5}{16}\).
- To find \(r\), compute the square root of \( \frac{5}{16} \), which results in \( \frac{\sqrt{5}}{4} \).
Remember, the radius extends from the center to any point on the circle, representing the circle's constant distance from the center.
