Intersection of a line and circle

AI Thread Summary
To find the value of a when the product m*n is minimized at the intersection of the line x+y=2a-1 and the circle x²+y²=a²+3a-3, the first step is to square the linear equation. This leads to the relationship (x+y)² = x² + y² + 2xy, allowing for the elimination of x²+y² from the equations. By substituting this into the circle's equation, xy can be expressed as a function of a. The resulting expression simplifies the problem, making it straightforward to determine the minimum value of m*n. Ultimately, solving this will yield the required value of a.
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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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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.
 

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