Maths Olympiad Prep

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Geometry Difficulty 4.6 AIME Find the answer

Find the shortest distance from the line 3x+4y=253 x+4 y=25 to the circle x2+y2=6x−8yx^{2}+y^{2}=6 x-8 y.

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

The circle is (x−3)2+(y+4)2=52(x-3)^{2}+(y+4)^{2}=5^{2}. The center (3,−4)(3,-4) is a distance of ∣3⋅3+4⋅−4−25∣32+42=325 \frac{|3 \cdot 3+4 \cdot-4-25|}{\sqrt{3^{2}+4^{2}}}=\frac{32}{5} from the line, so we subtract 5 for the radius of the circle and get 7/57 / 5.

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