Values such that the system of equations has only one solution

How many different values of a for which the system of equations

and has only one solution?

Re: Values such that the system of equations has only one solution

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**shiny718** How many different values of a for which the system of equations

and

has only one solution?

From the second equation, we have . Substituting into the first equation gives

For there to only be one solution, the discriminant must be 0, so

Checking the discriminant of this quadratic tells us how many values of a satisfy this equation, and therefore tell us how many values of a will give one solution to the original set of equations...

Since this is positive, there are two values of a which will satisfy the set of equations.

Re: Values such that the system of equations has only one solution

Another method. The first equation is that of a circle, which you can verify by representing it in the form (y - y₀)² + x² = r². How many lines with slope -1 touch this circle?

Re: Values such that the system of equations has only one solution

Thanks a lot! :) Thank you for making the solution really clear and simple. :)