Math CentralQuandaries & Queries


Question from Jamie, a student:

Use the Greek method to find an equation of the tangent line to the circle
x^2+y^2-4x+6y+4=0 at the points (3,2square root 2-3.

Hi Jamie,

I don't know what you mean by the Greek method but I can show you how I would solve this problem.

First rewrite the equation as

\[\left[ x^2 - 4x\right] + \left[y^2 + 6y \right] = -4.\]

Next I would complete the square for each of the expressions in square brackets to get an expression of the form

\[(x - a)^2 + (y - b)^2 = r^{2}.\]

This is an expression for the circle with center $(a ,b)$ and radius $r.$ Thus you know the coordinates of the center of the circle.

To complete this problem use that fact that for a circle with center $O$ and a point $P$ on the circumference of the circle, the line segment from $O$ to $P$ is perpendicular to the tangent to the circle at $P.$

Write back if you have problems completing the problem. Tell us what you have done and we will try to help,

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