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Standard Equation of a Circle

Learn about Standard Equation of a Circle as part of JEE Mathematics Mastery - Calculus and Algebra

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Mastering the Standard Equation of a Circle

Welcome to this module on the standard equation of a circle! Understanding this fundamental concept is crucial for success in coordinate geometry, especially for competitive exams like JEE. We'll break down the equation, its components, and how to use it effectively.

What is a Circle?

A circle is defined as the set of all points in a plane that are equidistant from a fixed point. This fixed point is called the center of the circle, and the constant distance is called the radius.

Deriving the Standard Equation

The distance formula is the key to the circle's equation.

Imagine a circle with its center at (h, k) and a radius 'r'. Any point (x, y) on the circle is exactly 'r' distance away from the center. We can use the distance formula to express this relationship.

Let the center of the circle be denoted by C(h, k) and any point on the circle be P(x, y). The radius of the circle is 'r'. According to the definition of a circle, the distance between the center C and any point P on the circle is equal to the radius 'r'.

Using the distance formula between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2), which is (x2−x1)2+(y2−y1)2\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}, we can find the distance CP:

CP=(x−h)2+(y−k)2CP = \sqrt{(x - h)^2 + (y - k)^2}

Since CP=rCP = r, we have:

(x−h)2+(y−k)2=r\sqrt{(x - h)^2 + (y - k)^2} = r

Squaring both sides to eliminate the square root, we get the standard equation of a circle:

(x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2

Understanding the Components

The standard equation (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2 tells us two crucial pieces of information about the circle:

ComponentMeaning in the Equation
Center (h, k)The coordinates (h, k) represent the location of the circle's center in the Cartesian plane.
Radius (r)The value 'r' is the radius of the circle. Note that the equation uses r2r^2, so you'll need to take the square root to find the actual radius.

Special Case: Circle Centered at the Origin

When the center of the circle is at the origin (0, 0), the values of h and k are both 0. Substituting these into the standard equation, we get a simplified form:

Origin-centered circles have a simpler equation.

If h=0 and k=0, the equation becomes x2+y2=r2x^2 + y^2 = r^2. This is the standard form for a circle whose center is at the origin.

For a circle centered at the origin (0, 0) with radius 'r', the standard equation (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2 becomes:

(x−0)2+(y−0)2=r2(x - 0)^2 + (y - 0)^2 = r^2

Which simplifies to:

x2+y2=r2x^2 + y^2 = r^2

This form is very common and easy to recognize.

Applying the Standard Equation

Let's look at how to use this equation. If you are given the center and radius, you can write the equation. Conversely, if you are given the equation, you can identify the center and radius.

What is the standard equation of a circle with center (h, k) and radius r?

(x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2

What does the value 'h' represent in the standard equation of a circle?

The x-coordinate of the circle's center.

If a circle's equation is x2+y2=25x^2 + y^2 = 25, what is its radius?

The radius is 5 (since r2=25r^2 = 25, r=25=5r = \sqrt{25} = 5).

Example Problem

Find the standard equation of a circle with center (-2, 3) and radius 4.

To find the standard equation, we substitute the given center coordinates (h, k) = (-2, 3) and the radius r = 4 into the formula (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2.

Here, h = -2 and k = 3. So, (x−(−2))2+(y−3)2=42(x - (-2))^2 + (y - 3)^2 = 4^2.

This simplifies to (x+2)2+(y−3)2=16(x + 2)^2 + (y - 3)^2 = 16. This equation represents a circle shifted 2 units to the left and 3 units up from the origin, with a radius of 4.

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Key Takeaways

The standard equation of a circle is a powerful tool for understanding and manipulating circles in the coordinate plane. Remember its form and how to extract the center and radius. Practice converting between the standard form and other forms to build your proficiency.

Learning Resources

Standard Equation of a Circle - Math is Fun(documentation)

Provides a clear explanation of the standard equation of a circle with interactive examples and definitions.

Equation of a Circle - Khan Academy(video)

A video tutorial explaining the derivation and components of the standard equation of a circle.

Circles: Standard Equation - Purplemath(documentation)

Detailed explanation of the standard equation of a circle, including how to graph circles from their equations.

Coordinate Geometry: Circles - BYJU'S(blog)

Covers various aspects of circles in coordinate geometry, including the standard equation and related properties.

The Standard Equation of a Circle - Varsity Tutors(documentation)

Explains the standard form of a circle's equation and provides examples of finding the equation given center and radius.

Circle Equation Explained - YouTube (Mathantics)(video)

An accessible video that breaks down the concept of the circle's equation in a visual and easy-to-understand manner.

Coordinate Geometry - Circles - Toppr(blog)

Offers comprehensive notes on circles in coordinate geometry, including the standard equation and its applications.

Equation of a Circle - Brilliant.org(documentation)

A wiki-style explanation of the circle's equation, focusing on its geometric interpretation and algebraic form.

JEE Mathematics: Coordinate Geometry - Circles - Vedantu(blog)

Content specifically tailored for JEE preparation, covering circles and their equations with solved examples.

Circle (mathematics) - Wikipedia(wikipedia)

A detailed overview of circles in mathematics, including their geometric properties and various algebraic representations.