1. Let \(k\) be a positive real number. The line \(x + y = k\) and the circle \(x^2 + y^2 = k\) are drawn. Find \(k\) so that the line is tangent to the circle.
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2. A circle passes through the points \((-2,0)\), \((2,0)\), and \((3,2)\). Find the center of the circle. Enter your answer as an ordered pair.
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Thank you!
+658 One question per post please.
1. This is mostly geometry believe it or not.
The value of K in the circle equation would be the blue line squared. Or in other words, the blue line is \(\sqrt{k}\)
The value of BC in the tangent equation would just be \(k\)
We know ABC is a 45 - 45 - 90 triangle (If the coefficients of x and y are 1 when graphed in standard form for the red line). The blue line is the perpendicular bisector.
Based on pythagorean theorem: \(2 *\sqrt{k}^2 = k^2\) (two legs are \(\sqrt{k}\) and hypotenuse is BC)
\(2k=k^2\)
\(k(k-2)=0\)
\(\boxed{k=2}\)
