Let \(\displaystyle{a}\in{\mathbb{{{R}}}}\). The number of distinct solutions

Pasegeabe85xy

Pasegeabe85xy

Answered question

2022-03-21

Let aR. The number of distinct solutions (x, y) that satisfy the system of equations (xa)2+y2=1 and x2=y2 can only be ?.

Answer & Explanation

Cody Hart

Cody Hart

Beginner2022-03-22Added 11 answers

I will say that (xa)2+y2=1
is the equation of a circle of radius r=1 centered at x=a. The additional constraint that x2=y2 implies that y=±x which are two lines of y-intercept 0 and slope ±1. Now depending on the value of a, the circle can be quite far from the origin where it will not intersect the lines (a.k.a. no solutions). It can also be close to the origin where it can have either 2 solutions or 4 solutions. Again, the answer depends heavily on the value of a.

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