What are the Intersection Points of a Circle and a Line?

In summary, the problem is to determine the values of k for which the line y = x + k intersects the circle x² + y² = 25 in one, two, or no points. The equations x² + y² = 25 and y = x + k are set equal to each other to find values of x and y that satisfy both equations. The resulting quadratic equation for x will have two distinct solutions when the line intersects the circle in two points.
  • #1
pokemeharder
18
0

Homework Statement


I'm completely stuck on this question
I'll write the whole thing down

Consider the circle x² + y² = 25 and the line y = x + k. where k is any real number. Determine the values of k for which the line will intersect the circle in one, two or no points.

PLEEEASE help me
Thanks


Homework Equations





The Attempt at a Solution

 
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  • #2
Didn't you ask a question almost exactly like this the other day? What have you tried so far?
 
  • #3
Points, (x,y), at which two curves intersect are x, y values that satisfy both equations. You are trying to find x and y that satisfy both x2+ y2= 25 and y= x+ k. If y= x+ k then the first equation becomes x2+ (x+ k)2= 25. That's a quadratic equation for x. When does a quadratic equation have two distinct solutions?
 
  • #4
nvm i got it yesterday lol
thanx anyways
 

1. What is a circle-line intersection?

A circle-line intersection is the point(s) where a circle and a straight line intersect. This can be visualized as the point(s) where the circle and line touch or cross each other.

2. How do you find the coordinates of a circle-line intersection?

The coordinates of a circle-line intersection can be found by solving the equations of the circle and line simultaneously. This can be done using algebraic methods or graphically by plotting the equations on a coordinate plane.

3. How many circle-line intersections are possible?

There can be a maximum of two circle-line intersections. These intersections can either be real (where the circle and line actually intersect) or imaginary (where the equations have no real solutions).

4. Can a circle and line intersect at multiple points?

No, a circle and line can only intersect at a maximum of two points. This is because a circle is a closed shape and cannot intersect with a line at more than two points.

5. How are circle-line intersections used in real-world applications?

Circle-line intersections are used in many fields such as geometry, physics, and engineering. They can be used to calculate the position of objects in motion, determine the intersection of roads or paths, and create accurate maps and diagrams. They also have applications in computer graphics and animation.

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