What is the Radius of a Circle Whose Equation is X2+Y2+8X−6Y+21=0? 2 Units 3 Units 4 Units 5 Units
When solving the given equation X^2 + Y^2 + 8X – 6Y + 21 = 0, you can rewrite it as (X + 4)^2 – 16 + (Y – 3)^2 – 9 + 21 = 0. This simplifies to (X + 4)^2 + (Y – 3)^2 = 4, and comparing it to the standard form (X – h)^2 + (Y – k)^2 = r^2, you can see that h = -4, k = 3, and r = √4, which gives r = 2.
Therefore, the radius of the given circle is 3 units.
Understanding Circle Equations
The equation of the circle is given by:
X2 + Y2 + 8X – 6Y + 21 = 0
To understand the radius of this circle, we need to interpret the coefficients:
- The coefficient of X2 is 1, which means the circle is not stretched or compressed in the X-direction.
- The coefficient of Y2 is 1, indicating that the circle is not stretched or compressed in the Y-direction.
- The coefficient of X (8) is twice the coefficient of X in X2, suggesting that the circle is shifted 4 units to the left in the X-direction.
- The coefficient of Y (-6) is twice the coefficient of Y in Y2, indicating that the circle is shifted 3 units upwards in the Y-direction.
- The constant term (21) determines the position of the circle in the coordinate plane.
To find the radius of the circle, we can complete the square:
X2 + 8X + Y2 – 6Y = -21
(X + 4)2 – 16 + (Y – 3)2 – 9 = -21
(X + 4)2 + (Y – 3)2 = 26
So, the radius of the circle is √26 units.
Calculating The Center Of The Circle
To calculate the center of a circle, we need to rewrite the given equation in standard form by completing the square. Let’s start by rearranging the equation as follows: (X^2 + 8X) + (Y^2 - 6Y) = -21. To complete the square for the X terms, we add (8/2)^2 = 16, and for the Y terms, we add (-6/2)^2 = 9. The equation becomes (X^2 + 8X + 16) + (Y^2 - 6Y + 9) = -21 + 16 + 9. Simplifying further, we have (X + 4)^2 + (Y - 3)^2 = 4. Now, we can identify the center point of the circle as (-4, 3) by comparing it with the standard form equation (X - h)^2 + (Y - k)^2 = r^2, where the center is given by (h, k).
Determining The Radius
To determine the radius of a circle with the equation X2+Y2+8X−6Y+21=0, we apply the distance formula. By simplifying the equation, we can find the radius. For the given equation, the radius is 4 units.
Solving For Different Radius Values
To calculate the radius of a circle with a given equation, we can complete the square. For the equation X2 + Y2 + 8X – 6Y + 21 = 0, the center of the circle can be found by rearranging the terms as (X + 4)2 + (Y – 3)2 = R2. For a radius of 2 units, we have R = 2. For a radius of 3 units, we have R = 3. For a radius of 4 units, we have R = 4. For a radius of 5 units, we have R = 5.
Conclusion
To summarize, the radius of a circle can be determined by analyzing its equation, such as the one provided: X^2+Y^2+8X−6Y+21=0. By properly rearranging the equation and identifying its center coordinates, we can easily calculate the radius. In this case, the radius of the circle is either 2 units, 3 units, 4 units, or 5 units, depending on certain conditions.
Understanding the equation’s components allows for a precise determination of the circle’s radius, which is essential in various mathematical and practical applications.