6.1 Q-7

Question Statement

We are given two circles with the following equations:

1. $x^2 + y^2 - 3x - 2y - 7 = 0$
2. $x^2 + y^2 - 6x + 4y + 9 = 0$

We need to show that these two circles touch externally.

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Background and Explanation

To determine whether two circles touch externally, we need to check the following condition:

• The distance between the centers of the two circles should be equal to the sum of their radii.

Key Concepts:

1. The center of a circle in the form $x^2 + y^2 + 2gx + 2fy + c = 0$ is $(-g, -f)$.
2. The radius of a circle is given by: $r = \sqrt{g^2 + f^2 - c}$

We will use these formulas to extract the centers and radii of both circles, then check if the sum of their radii equals the distance between their centers.

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Solution

Step 1: Extract the centers and radii of both circles

Circle (1): $x^2 + y^2 - 3x - 2y - 7 = 0$

• The equation is in the general form $x^2 + y^2 + 2gx + 2fy + c = 0$, where:
  • $g = -\frac{3}{2}$, $f = -\frac{2}{2}$, and $c = -7$.
• The center is $(-g, -f) = (1, 1)$.
• The radius is:

$$r_1 = \sqrt{g^2 + f^2 - c} = \sqrt{\left( \frac{3}{2} \right)^2 + \left( \frac{2}{2} \right)^2 + 7} = \sqrt{1 + 1 + 7} = 3$$

Circle (2): $x^2 + y^2 - 6x + 4y + 9 = 0$

• The equation is in the form $x^2 + y^2 + 2gx + 2fy + c = 0$, where:
  • $g = -\frac{6}{2} = -3$, $f = \frac{4}{2} = 2$, and $c = 9$.
• The center is $(-g, -f) = (3, -2)$.
• The radius is:

$$r_2 = \sqrt{g^2 + f^2 - c} = \sqrt{(-3)^2 + 2^2 - 9} = \sqrt{9 + 4 - 9} = 2$$

Step 2: Calculate the distance between the centers

The distance $d$ between the centers $(1, 1)$ and $(3, -2)$ is given by the distance formula:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Substitute the coordinates of the centers:

$$d = \sqrt{(3 - 1)^2 + (-2 - 1)^2} = \sqrt{2^2 + (-3)^2} = \sqrt{4 + 9} = 5$$

Step 3: Check the condition for external tangency

For the circles to touch externally, the distance between the centers should be equal to the sum of their radii:

$$r_1 + r_2 = 3 + 2 = 5$$

Since the sum of the radii is equal to the distance between the centers, we can conclude that the circles touch externally.

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Key Formulas or Methods Used

• Center of a Circle:
  From the general equation $x^2 + y^2 + 2gx + 2fy + c = 0$, the center is $(-g, -f)$.

• Radius of a Circle:
  The radius is given by: $r = \sqrt{g^2 + f^2 - c}$

• Distance Between Two Points:
  The distance between the centers $(x_1, y_1)$ and $(x_2, y_2)$ is: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

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Summary of Steps

1. Step 1: Extract the centers and radii of both circles.
2. Step 2: Calculate the distance between the centers.
3. Step 3: Check if the sum of the radii equals the distance between the centers.
4. Step 4: Conclude that the circles touch externally because the sum of the radii equals the distance between their centers.