Problem 60 - Algebra and Trigonometry

Question

55–62 ? Find an equation of the circle that satisfies the given conditions. Endpoints of a diameter are \(P(-1,3)\) and \(Q(7,-5)\)

Step-by-Step Solution

Answer

The equation of the circle is \((x - 3)^2 + (y + 1)^2 = 32\).

Step 1: Find the Midpoint

The midpoint of a diameter of a circle is the center of the circle. To find the midpoint between points \(P(-1, 3)\) and \(Q(7, -5)\), use the midpoint formula: \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \). This gives: \( \left( \frac{-1 + 7}{2}, \frac{3 - 5}{2} \right) = (3, -1) \). So, the center of the circle is \((3, -1)\).

Step 2: Calculate the Radius

The radius of the circle is half the length of the diameter. Calculate the length of the diameter using the distance formula: \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Substituting \((-1, 3)\) and \((7, -5)\), we get \( \sqrt{(7 - (-1))^2 + (-5 - 3)^2} = \sqrt{8^2 + (-8)^2} = \sqrt{64 + 64} = \sqrt{128} = 8 \sqrt{2} \). The radius is thus \(\frac{8\sqrt{2}}{2} = 4\sqrt{2}\).

Step 3: Write the Equation of the Circle

The equation of a circle with center \( (h, k) \) and radius \( r \) is \((x - h)^2 + (y - k)^2 = r^2\). With center \((3, -1)\) and radius \(4\sqrt{2}\), the equation becomes: \((x - 3)^2 + (y + 1)^2 = (4\sqrt{2})^2 = 32\).

Key Concepts

Midpoint Formula

The midpoint formula is a nifty tool used to find the exact center between two points on a plane. In this case, we have the endpoints of a circle's diameter, which are points \( P(-1, 3) \) and \( Q(7, -5) \). The midpoint is crucial because it also represents the center of the circle.

When applying the midpoint formula, you calculate the average of the \( x \)-coordinates and the \( y \)-coordinates separately. The formula is as follows:

• Midpoint \((x, y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\)

For our points \( P \) and \( Q \), plugging them into the equation gives us:

• \( \left( \frac{-1 + 7}{2}, \frac{3 - 5}{2} \right) = (3, -1) \)

Thus, the center of the circle is at \( (3, -1) \). This step is fundamental because any circle's equation will revolve around its center.

Distance Formula

To find the length of the circle's diameter, we use the distance formula. This calculation is essential to determine how long the straight line segment connecting points \( P \) and \( Q \) is. The distance gives us the diameter of the circle.

The distance formula is expressed as:

• \[ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

For points \( P(-1, 3) \) and \( Q(7, -5) \), substitute the coordinates into the formula:

• \[ \sqrt{(7 - (-1))^2 + (-5 - 3)^2} = \sqrt{64 + 64} = \sqrt{128} = 8\sqrt{2} \]

This result tells us that the diameter of the circle is \( 8\sqrt{2} \). Knowing the diameter is a pivotal step toward unearthing the circle's radius.

Radius Calculation

The radius of the circle is half the length of its diameter. This value is needed to complete the equation of the circle.

From the distance formula, we previously found the diameter to be \( 8\sqrt{2} \). Therefore, the radius \( r \) is:

• \( r = \frac{8\sqrt{2}}{2} = 4\sqrt{2} \)

The radius acts as one of the crucial variables in forming the equation of the circle. With the radius in hand, we can now write the full equation for this specific circle!

Center of Circle

The center of the circle is derived using the midpoint formula, as explained earlier. This central point \( (h, k) \) is critical because it's part of the circle's standard equation format.Using our previously found midpoint, the center of this circle is \( (3, -1) \). This point allows us to place the circle in a coordinated plane and aligns with the formation of the circle's equation.

The general formula for the equation of a circle is:

• \[ (x - h)^2 + (y - k)^2 = r^2 \]

With a center at \((3, -1)\) and a radius of \(4\sqrt{2}\), we substitute these values into the formula to get:

• \( (x - 3)^2 + (y + 1)^2 = 32 \)

This equation completely defines our circle based on the given diameter endpoints. Understanding each piece and how to derive them is crucial, as every piece of information builds on each other.