Find The Angle Between The Pair Of Lines Joining The Origin To The Points Of Intersection Of The Equations X 2 + Y 2 − 2 X − 2 Y − 8 = 0 X^2 + Y^2 - 2x - 2y - 8 = 0 X 2 + Y 2 − 2 X − 2 Y − 8 = 0 And X + Y − 1 = 0 X + Y - 1 = 0 X + Y − 1 = 0 .
Introduction
In this problem, we are given two equations of circles and a line. We need to find the points of intersection of the two circles and the line, and then find the angle between the pair of lines joining the origin to these points of intersection. This problem involves solving a system of equations, finding the points of intersection, and then using trigonometry to find the angle between the lines.
Step 1: Solve the System of Equations
To find the points of intersection, we need to solve the system of equations formed by the two circle equations and the line equation. We can start by solving the line equation for y:
Now, substitute this expression for y into the circle equation:
Expand and simplify the equation:
Combine like terms:
Step 2: Solve the Quadratic Equation
Now, we need to solve the quadratic equation:
We can use the quadratic formula to solve for x:
In this case, a = 2, b = -4, and c = -9. Plugging these values into the formula, we get:
Simplify the expression under the square root:
Step 3: Find the Corresponding y-Values
Now that we have the x-values, we can find the corresponding y-values by substituting the x-values into the equation of the line:
For the first x-value, we get:
For the second x-value, we get:
Step 4: Find the Angle Between the Pair of Lines
Now that we have the points of intersection, we can find the angle between the pair of lines joining the origin to these points. We can use the law of cosines to find the angle:
In this case, a and b are the distances from the origin to the points of intersection, and c is the distance between the points of intersection. We can find these distances using the distance formula:
For the first point, we get:
For the second point, we get:
The distance between the points of intersection is:
Now, we can plug these values into the law of cosines:
Simplify the expression:
This means that the angle between the pair of lines is 180 degrees.
Conclusion
In this problem, we found the points of intersection of the two circle equations and the line equation, and then found the angle between the pair of lines joining the origin to these points of intersection. We used the law of cosines to find the angle, and found that it is 180 degrees. This problem involved solving a system of equations, finding the points of intersection, and then using trigonometry to find the angle between the lines.
Q&A
Q: What is the main concept behind finding the angle between the pair of lines joining the origin to the points of intersection?
A: The main concept behind finding the angle between the pair of lines joining the origin to the points of intersection is to use the law of cosines. The law of cosines states that for any triangle with sides of length a, b, and c, and angle θ opposite side c, the following equation holds:
Q: How do we find the points of intersection of the two circle equations and the line equation?
A: To find the points of intersection, we need to solve the system of equations formed by the two circle equations and the line equation. We can start by solving the line equation for y:
Now, substitute this expression for y into the circle equation:
Expand and simplify the equation:
Combine like terms:
We can then solve the quadratic equation using the quadratic formula:
In this case, a = 2, b = -4, and c = -9. Plugging these values into the formula, we get:
Simplify the expression under the square root:
Q: How do we find the corresponding y-values?
A: Now that we have the x-values, we can find the corresponding y-values by substituting the x-values into the equation of the line:
For the first x-value, we get:
For the second x-value, we get:
Q: How do we find the angle between the pair of lines?
A: Now that we have the points of intersection, we can find the angle between the pair of lines joining the origin to these points. We can use the law of cosines to find the angle:
In this case, a and b are the distances from the origin to the points of intersection, and c is the distance between the points of intersection. We can find these distances using the distance formula:
For the first point, we get:
For the second point, we get:
The distance between the points of intersection is:
Now, we can plug these values into the law of cosines:
Simplify the expression:
This means that the angle between the pair of lines is 180 degrees.
Q: What is the significance of finding the angle between the pair of lines?
A: Finding the angle between the pair of lines is significant in various fields such as physics, engineering, and computer science. It can be used to determine the orientation of objects, the trajectory of projectiles, and the behavior of electrical circuits.
Q: How can we apply the concept of finding the angle between the pair of lines in real-world scenarios?
A: The concept of finding the angle between the pair of lines can be applied in various real-world scenarios such as:
- Determining the orientation of buildings and structures
- Calculating the trajectory of projectiles and missiles
- Analyzing the behavior of electrical circuits and networks
- Designing and optimizing mechanical systems and mechanisms
Q: What are the limitations of the concept of finding the angle between the pair of lines?
A: The concept of finding the angle between the pair of lines has some limitations such as:
- It assumes that the lines are straight and not curved
- It assumes that the lines are not intersecting or parallel
- It assumes that the lines are not affected by external factors such as gravity or friction
Q: How can we overcome the limitations of the concept of finding the angle between the pair of lines?
A: To overcome the limitations of the concept of finding the angle between the pair of lines, we can use more advanced mathematical techniques such as:
- Using differential geometry to analyze curved lines
- Using vector calculus to analyze intersecting or parallel lines
- Using numerical methods to analyze lines affected by external factors such as gravity or friction