Understanding the Intersection of a Circle and a Line

In this article, we will explore how to find the intersection points of the circle defined by the equation (x^2 y^2 25) and the line defined by the equation (4x 3y 12). Additionally, we will calculate the length of the intersection chord formed by these points.

Step-by-Step Solution

Let's start with the given circle and line equations:

Circle: (x^2 y^2 25)

Line: (4x 3y 12)

To find the intersection points, we need to substitute the line equation into the circle equation. First, we solve the line equation for x:

(4x 3y 12 implies 4x 12 - 3y implies x frac{12 - 3y}{4})

Substituting into the Circle Equation

Now, we substitute (x frac{12 - 3y}{4}) into the circle equation:

(x^2 y^2 25 implies left(frac{12 - 3y}{4}right)^2 y^2 25)

Simplifying the above equation:

(frac{(12 - 3y)^2}{16} y^2 25 implies (12 - 3y)^2 16y^2 400)

(implies 144 - 72y 9y^2 16y^2 400 implies 25y^2 - 72y 144 400 implies 25y^2 - 72y - 256 0)

Solving the Quadratic Equation

We solve the quadratic equation (25y^2 - 72y - 256 0) using the quadratic formula:

(y frac{-b pm sqrt{b^2 - 4ac}}{2a})

Here, (a 25), (b -72), and (c -256).

(y frac{72 pm sqrt{72^2 - 4 times 25 times (-256)}}{2 times 25})

(y frac{72 pm sqrt{5184 25600}}{50} implies y frac{72 pm sqrt{30784}}{50})

(y frac{72 pm 175.4537}{50})

Therefore, the solutions for (y) are:

(y_1 frac{72 175.4537}{50} approx 4.949)

(y_2 frac{72 - 175.4537}{50} approx -2.069)

Finding Corresponding x Values

Now, we find the corresponding (x) values for (y_1) and (y_2):

For (y_1 4.949):

(x frac{12 - 3 times 4.949}{4} approx frac{12 - 14.847}{4} approx -0.712)

For (y_2 -2.069):

(x frac{12 - 3 times (-2.069)}{4} approx frac{12 6.207}{4} approx 4.552)

Calculating the Length of the Chord

Now, we calculate the distance between the points ((-0.712, 4.949)) and ((4.552, -2.069)):

(text{Chord length} sqrt{(4.552 - (-0.712))^2 (-2.069 - 4.949)^2} sqrt{(4.552 0.712)^2 (-2.069 - 4.949)^2})

( sqrt{(5.264)^2 (-6.998)^2} sqrt{27.7056 48.9760} sqrt{76.6816} approx 8.757)

Therefore, the length of the intersection chord is approximately 8.757 units.

Conclusion

This method provides a step-by-step guide to solving the problem of finding the intersection points of a circle and a line, and calculating the length of the intersection chord. Understanding these concepts is fundamental in geometry and analytic algebra.

Related Keywords

circle equation intersection point chord length