Math CentralQuandaries & Queries


Question from Mohammad, a student:

Find the equation of circle passing through the origin and the points (a,b) and (b,a).
Find the length of chords that it cuts off from the axes.

Hi Mohammad,

The key observation here is that the center of your circle must be on the line $y=x$ (that goes through the origin and through the midpoint of $(a,b)$ and $(b,a)$). A typical point on this line is $(c, c),$ so your first job is to find the equation of the circle with center $(c, c)$ that passes through the origin — you can almost do that in your head! (Did you get $x^2 + y^2 -2c(x+y) = 0 ?$)

To find out what $c$ is in terms of $a$ and $b,$ simply replace $x$ by $a$ and $y$ by $b$ in your equation of the circle, then solve for $c.$ You end up with the equation of a circle with $c$ replaced by an expression involving $a$ and $b.$

To find the chord lengths plug $y=0$ into the equation of the circle and solve for $x.$


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