# MATH / 

task_id: 78015003-8cbb-57f7-9bb3-8d80ff277300
task_key: intermediate~5falgebra--test--78015003-8cbb-57f7-9bb3-8d80ff277300
task_revision_id: 1

{"level":"Level 5","problem":"Let $A$ and $B$ be two points lying on the parabola $y^2 = 4x$ in the first quadrant.  The circle with diameter $\\overline{AB}$ has radius $r,$ and is tangent to the $x$-axis.  Find the slope of line $AB$ in terms of $r.$\n\n[asy]\nunitsize(0.4 cm);\n\npath parab = (16,-8);\nreal y;\npair A, B, O;\nreal a, b, r;\n\na = (10 + 2*sqrt(5))/5;\nb = (10 - 2*sqrt(5))/5;\nA = (a^2,2*a);\nB = (b^2,2*b);\nO = (A + B)/2;\nr = a + b;\n\nfor (y = -8; y <= 8; y = y + 0.2) {\n  parab = parab--(y^2/4,y);\n}\n\ndraw(parab,red);\ndraw((-2,0)--(16,0));\ndraw((0,-8)--(0,8));\ndraw(Circle(O,r));\ndraw(A--B);\n\ndot(\"$A$\", A, N);\ndot(\"$B$\", B, W);\n[/asy]","type":"Intermediate Algebra"}

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