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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!

May 3, 2020

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2) The center of the circle is (0, 7/4).

May 3, 2020
#2
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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)

$$2k=k^2$$

$$k(k-2)=0$$

$$\boxed{k=2}$$

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May 3, 2020