6.1 Q-5

Question Statement

Find the equation of a circle with the following conditions:

The radius is $a$ .
The circle lies in the second quadrant and is tangent to both the $x$ - and $y$ -axes.

Background and Explanation

To solve this problem, we need to recall the general equation of a circle:

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

Where:

$(h, k)$ is the center of the circle.
$r$ is the radius.

The key insight here is that the circle is tangent to both axes and lies in the second quadrant. This means that the center of the circle must be at a point where both the $x$ -coordinate and $y$ -coordinate are negative. Specifically, the center will be at $(-a, a)$ , because the distance from the center to the axes is equal to the radius $a$ , and the circle is tangent to both axes.

Solution

Step 1: Understanding the Center and Radius

Since the circle is tangent to both axes and lies in the second quadrant, the center must be at $(-a, a)$ and the radius is $a$ .

Step 2: Using the General Equation of a Circle

The general equation of a circle is:

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

Substituting $h = -a$ , $k = a$ , and $r = a$ , the equation becomes:

(x + a)^2 + (y - a)^2 = a^2 \tag{1}

Step 3: Expanding the Equation

Next, expand both terms in the equation:

(x + a)^2 = x^2 + 2ax + a^2

(y - a)^2 = y^2 - 2ay + a^2

Substitute these into equation (1):

x^2 + 2ax + a^2 + y^2 - 2ay + a^2 = a^2

Step 4: Simplifying the Equation

Combine like terms:

x^2 + y^2 + 2ax - 2ay + 2a^2 = a^2

Now, move the constant $a^2$ to the right side:

x^2 + y^2 + 2ax - 2ay + a^2 = 0

Thus, the equation of the circle is:

x^2 + y^2 + 2ax - 2ay + a^2 = 0

Key Formulas or Methods Used

General Equation of a Circle:
$(x - h)^2 + (y - k)^2 = r^2$
Where $(h, k)$ is the center and $r$ is the radius.
Expansion of Squared Terms:
Expand $(x + a)^2$ and $(y - a)^2$ to get the terms involving $x$ , $y$ , and constants.
Simplification:
Combine like terms to obtain the final equation of the circle.

Summary of Steps

Step 1: Identify the center and radius of the circle based on the given conditions.
Step 2: Substitute the values for $h$ , $k$ , and $r$ into the general equation of a circle.
Step 3: Expand both squared terms and simplify.
Step 4: Simplify the equation further to obtain the final form.