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What is the radius of a circle whose equation is x2+y2+8x−6y+21=0? 2 units 3 units 4 units 5 units

What is the radius of a circle whose equation is x2+y2+8x−6y+21=0? 2 units 3 units 4 units 5 units – A circle may be algebraically described using the circle equation, which considers the radius and center lengths. The formulae used to determine a circle’s area or perimeter differ from the equation of a circle. Numerous circle problems in coordinate geometry employ this equation.

We require the circle’s equation to depict a circle on the Cartesian plane. If we know a circle’s radius and center, we can draw the circle on paper. Similarly, if we know the coordinates of the center and radius of a circle, we may mark it on the Cartesian plane. There are several methods to symbolize a circle:

general form

standard form

parametric form

polar form

Let’s learn the circle equation and its various shapes with graphs and solved examples.

What is the Circle Equation?

What is the Circle Equation?

The location of a circle in the Cartesian plane is represented by its equation. Writing down the circle’s equation is possible if we know the radius and center’s coordinates. The circle equation represents every point on the circle’s circumference.

The positions of points whose separation from a fixed point is a fixed value are represented by a circle. The circle’s radius is its constant value; this fixed point is known as its center. Equation standard for a center circle

(X1, y1) and r radius (X−X1)2+(y−y1)2=R.2.

How do you find the equation of a circle?

There are many different ways to represent the circle equation, depending on the circle’s position in the Cartesian plane. We examined the forms that represent the circle equation for given coordinates of the center of a circle. Some exceptional cases depend on the circle’s location in the coordinate plane. Let’s learn how to find the circle equation in general and for these exceptional cases.

Different Forms of the Circle Equation

A circle equation represents the location of a circle in the Cartesian plane. Given the radius and center lengths, a circle may be sketched on paper. We may draw the circle on the Cartesian plane after utilizing the circle equation to determine the circle’s radius and center coordinates. There are several ways to display a circle’s equation,

Here, let’s look in detail at the two general forms of the circle equation and the standard form of the circle equation, polar and parametric forms.

General Equation of a Circle

An equation in the general form is x2 + y2 + 2gx + 2fy + c = 0. The circle’s radius and center coordinates may be found using this generic form, where g, f, and c are constants. Finding significant features about any circle is challenging when using the broad form of a circle equation, in contrast to the standard form, which is more straightforward to comprehend. As a result, we will quickly convert from the general form to the traditional form using the square completion technique.

What is the Equation of a Circle?

What is the Equation of a Circle?

A circle is a closed curve drawn from a fixed point called the center, where all points on the curve are the same distance from the center point of the center. The equation of a circle with center (h, k) and radius r is given as follows:

(x-h)2 + (y-k)2 = r2

It is the standard form of the equation. Therefore, if we know the coordinates of the center of the circle and its radius, we can easily find its equation

Example: Let’s assume that point (1,2) is the center of the circle, and its radius is 4 cm. Then the equation of this circle will be

(x-1)2+(y-2)2 = 42

(x2−2x+1)+(y2−4y+4) =16

x2+y2−2x−4y-11 = 0

  • Function or Not

We know that the question arises whether the circle is a function or not. The circle is not a function. A function is defined by the fact that each value in its domain is precisely related to a point in the common domain, but a line passing through the circle intersects this line at two points on the surface.

The mathematical way to describe a circle is as an equation. Here, the equation of the circle is given with examples in all forms, such as general form and standard form.

Equation of a Circle Where the Center is the Origin

Consider an arbitrary point P(x, y) on the circle. Let ‘A’ be the radius of the circle equal to OP.

We know that the distance between point (x, y) and origin (0,0) can be found using the distance formula equal to:

√[x2+ y2]= a

  • Therefore, the equation of a circle whose center is the origin is

x2+y2= a2

  • Here “a” is the radius of the circle.
  1. Alternative Method

Let’s derive it another way. Let’s assume that (x,y) is a point on a circle and the center of the circle is at the origin (0,0). If we draw a perpendicular from the point (x,y) to the x-axis, we get a right triangle where the circle’s radius is the hypotenuse. The triangle’s base is the distance along the x-axis, and the height is the distance along the y-axis. So, applying the Pythagorean theorem here, we get:

x2+y2 = radius2

Equation of a Circle When the Center Is Not the Origin

Equation of a Circle When the Center Is Not the Origin

Let C(h, k) be the circle’s center and P(x, y) be any point on the circle.

  1. Therefore, the radius of the circle is CP

Using the distance formula,

(x-h)2 + (y-k)2 = CP2

Let the radius be ‘a’.

  1. Therefore, the equation of the circle with center (h, k) and radius ‘a’ is

(x-h)2+(y-k)2 = a

This is called the standard form of the circle equation.

General Form of Circle Equation

The general equation of any circle is represented as:

x2 + y2 + 2gx + 2fy + c = 0 for all g, f and c values.

Adding g2 + f2 to both sides of the equation gives

x2 + 2gx + g2+ y2 + 2fy + f2= g2 + f2 − c ………………(1)

Substituting (x+g)2 = x2+ 2gx + g2 and (y+f)2 =y2 + 2fy + f2 into equation (1), we get:

(x+g)2+ (y+f)2 = g2 + f2−c …………….(2

If we compare (2) with (x−h)2 + (y−k)2 = a2, where (h, k) is the center of the circle and ‘a’ is the radius.

h=−g, k=−f

a2 = g2+ f2−c

Then,

x2 + y2 + 2gx + 2fy + c = 0 represents the circle with center (−g,−f) and radius equal to a2 = g2 + f2− c.

If g2 + f2 > c, the circle’s radius is real.

If g2 + f2 = c, the circle’s radius is zero, which tells us that the circle is a point coinciding with the center. Such a circle is called a point circle.

If g2 + f2 <c, the circle’s radius becomes virtual. Therefore, it is a circle with an actual center and imaginary radius.

Conclusion

What is the radius of a circle whose equation is x2 + y2 + 8x – 6y + 21 = 0?

Solution:

Given circle equation x2 + y2 + 8x – 6y + 21=0

We have a standard equation of a circle,

x2 + y2 – 2ax – 2by + (a2 + b2 – m2) = 0 with centre(a, b) and radius “m”

To find the radius, we extract the “m” term

m = √(a2 +b2 ) — (a)

Compare the terms (a2 + b2 – m2) = 2

-2ax = 8x ⇒ a = -4

-2by = -6 ⇒ b =

Put a,b values in eq(a)

Radius = m = √(a2 +b2 ) =√((-4)2 +32)

= √(16 +9)

= √25 = 5

Radius = 5 units.

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