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Find the equation of the circles passing through \((-4,3)\) and touching the lines \(x+y=2\) and \(x-y=2\)

Short Answer

Expert verified
The equations of the circles are \((x-3)^2 + (y-1)^2 = 2^2\) and \((x+1)^2 + (y-1)^2 = 2^2\).
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Step 1: Equation of Tangent Lines

First, find the slopes of the two lines. Since they are tangent to the circle, the radii that connect the center of the circle to the points of tangency are going to be perpendicular to the tangent lines. Therefore, the slopes of these radii will be the opposite reciprocals of the slopes of the tangent lines. The slopes of the two lines given are \(1\) and \(-1\), so the slopes of the radii are \(-1\) and \(1\), respectively.

Step 2: Equation of Circle

If \((h,k)\) represents the center of the circle and \(r\) the radius, the equation of a circle is \((x-h)^2 + (y-k)^2 = r^2\). Because we have the point \((-4,3)\) on the circle, substituting these values yields \((h+4)^2 + (k-3)^2 = r^2\) (equation 1). Due to tangency, the distance from the center to each line also equals the radius \(r\). The distance from \((h,k)\) to the line \(ax+by+c=0\) is \(\frac{|ah+bk+c|}{\sqrt{a^2+b^2}}\). So substituting for each line gives \(r = \frac{|h+k-2|}{\sqrt{2}}\) (equation 2) and \(r = \frac{|h-k-2|}{\sqrt{2}}\) (equation 3).

Step 3: Solve the Equations

Setting equation 2 equal to equation 3 gives an equation in terms of \(h\) and \(k\). Solving this yields \(k = 1\). Substituting \(k\) and the given point \((-4,3)\) into equation 1, and solving the resulting equation will give the possible values for \(h\) and finally \(r\).

Step 4: Write the Final Equations

Substitute the values of \(h\), \(k\), and \(r\) into the standard form for the circle's equation. This will provide the equations of the circles that pass through the given point and touch the two lines.

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