Given: 2 lines containing the diameter of a circle and a point lying on this circle; Find: the equation of this circle

Question

The lines $ y = \frac{4}{3}x - \frac{5}{3} $ and $ y = \frac{-4}{3}x - \frac{13}{3} $ each contain diameters of a circle. and the point $ (-5, 0) $ is also on that circle.

Find the equation of this circle.

Answer 1

Prerequisites

---

Given

• We have 2 lines with given equations (see below).

  • $ y = \frac{4}{3}x - \frac{5}{3} $
  • $ y = \frac{-4}{3}x - \frac{13}{3} $

• These lines each contain a diameter of the circle-in-question.

• We have a point with the given coordinate (see below).

  • $ P = (-5, 0) $

• This point is on the circle in question.

---

Problem

Find the equation of the circle-in-question.

Here is our progress:

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

---

Solution

---

Step 1

Let us find the center of the circle-in-question.

Note that diameters of circles go through the center of a circle.

We can find the intersection of the two given lines.

Let us find the x-component of the center of the circle-in-question.

$ \frac{4}{3}x - \frac{5}{3} = \frac{-4}{3}x - \frac{13}{3} $

$ 4x - 5 = -4x - 13 $

$ 8x = -8 $

$ x = -1 $

Let us find the y-component of the center of the circle-in-question.

$ y = \frac{4}{3}x - \frac{5}{3} = \frac{4}{3}(-1) - \frac{5}{3} = \frac{-4}{3} - \frac{5}{3} = \frac{-9}{3} = -3 $

So...

The center of the circle-in-question is $ Q = (-1, -3) $.

Here is our progress:

$ (x - (-1))^2 + (y - (-3))^2 = r^2 $

$ (x + 1)^2 + (y + 3)^2 = r^2 $

---

Step 2

Let us find the radius of the circle-in-question.

We know that the radius of the circle-in-question is the distance between the center of the circle-in-question and the given point that lies on the circle-in-question.

Let us find the distance between such points.

$ l = \sqrt{((-1)-(-5))^2+((-3)-(0))^2} = \sqrt{(4)^2+(-3)^2} = \sqrt{16+9} = \sqrt{25} = 5 $

So...

The length of the radius is $ r = 5 $

Here is our progress:

$ (x + 1)^2 + (y + 3)^2 = (5)^2 $ $ (x + 1)^2 + (y + 3)^2 = 25 $

---

Answer

---

The equation of the circle-in-question is:

$$ (x + 1)^2 + (y + 3)^2 = 25 $$