Intersection of a line and circle

In summary, the formula for finding the intersection point between a line and a circle is to set their equations equal to each other and solve the resulting quadratic equation. A line can only intersect a circle at a maximum of two points, and they are considered tangent if they touch at exactly one point. A line and a circle can also be parallel to each other, with a distance equal to the radius of the circle. The number of solutions for the intersection of a line and a circle can be zero, one, or two, depending on their position relative to each other.
  • #1
bosonhiggs
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Homework Statement



The line x+y=2a-1 intersects the circle x2+y2=a2+3a-3 at point (m,n). When m*n reaches its minimum value, what is the value of a?

Homework Equations



Equation One

x + y = 2a - 1

Equation Two

x2 + y 2 = a 2 + 3a - 3
 
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  • #2
Square your linear (first) equation, using the fact that (x+y)^2 = x^2 + y^2 + 2xy, at which point you can easily eliminate "x^2 + y^2" from the two equations. Now you have "xy" as a function of "a" and the rest should be easy.
 

1. What is the formula for finding the intersection point between a line and a circle?

The formula for finding the intersection point between a line and a circle is to set the equations for the line and the circle equal to each other. This will result in a quadratic equation, which can then be solved using the quadratic formula.

2. Can a line intersect a circle at more than two points?

No, a line can only intersect a circle at a maximum of two points. This is because a line can be considered as an infinite set of points, but a circle is a finite shape with a fixed radius.

3. How do you determine if a line and a circle are tangent to each other?

A line and a circle are tangent to each other if the line touches the circle at exactly one point. This means that the distance between the center of the circle and the point of tangency is equal to the radius of the circle.

4. Can a line and a circle be parallel to each other?

Yes, a line and a circle can be parallel to each other. This occurs when the line is outside of the circle and does not intersect it at any point. In this case, the distance between the line and the circle will be equal to the radius of the circle.

5. How many solutions are there for the intersection of a line and a circle?

There can be either zero, one, or two solutions for the intersection of a line and a circle. If the line is completely outside of the circle, there will be no intersection points. If the line is tangent to the circle, there will be one intersection point. And if the line intersects the circle at two points, there will be two intersection points.

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