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Tangent lines are drawn from the external point P(-2, 0) to the parabola y² = 8x. Find the radius/radii of the circle(s) that are simultaneously tangent to both of these tangent lines and to the chord of contact of the tangents with the parabola.
- 4
- 4*(sqrt(2) - 1)
- 4*sqrt(2)
- 4*sqrt(2) + 4
Correct answer: 4*(sqrt(2) - 1)
Solution
The two tangents from (-2,0) to y²=8x are y=x+2 and y=-x-2, with chord of contact x=2. A circle centered at (h,0) of radius r must satisfy r=|h+2|/sqrt(2) and r=|h-2|. Solving gives two valid circles: one with r=4(sqrt(2)-1) (center between the tangents and chord) and one with r=4(sqrt(2)+1) (center beyond the chord). Both 4(sqrt(2)-1) and 4*sqrt(2)+4 are valid answers.
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