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MATH / Let A and B be two points lying on the parabola y²=4x in the first quadrant. The circle with diameter (AB)¯ has radius r, and is tangent to the x-axis. Find the slope of line AB in…

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Answer published by the source. Consult the official source to check your work against its answer.

problem

Let AA and BB be two points lying on the parabola y2=4xy^2 = 4x in the first quadrant. The circle with diameter AB¯\overline{AB} has radius r,r, and is tangent to the xx-axis. Find the slope of line ABAB in terms of r.r. [asy] unitsize(0.4 cm); path parab = (16,-8); real y; pair A, B, O; real a, b, r; a = (10 + 2*sqrt(5))/5; b = (10 - 2*sqrt(5))/5; A = (a^2,2*a); B = (b^2,2*b); O = (A + B)/2; r = a + b; for (y = -8; y <= 8; y = y + 0.2) { parab = parab--(y^2/4,y); } draw(parab,red); draw((-2,0)--(16,0)); draw((0,-8)--(0,8)); draw(Circle(O,r)); draw(A--B); dot("AA", A, N); dot("BB", B, W); [/asy]
Plain-text mathematical notation (without MathML)
Let A and B be two points lying on the parabola y²=4x in the first quadrant.  The circle with diameter (AB)¯ has radius r, and is tangent to the x-axis.  Find the slope of line AB in terms of r.

[asy]
unitsize(0.4 cm);

path parab = (16,-8);
real y;
pair A, B, O;
real a, b, r;

a = (10 + 2*sqrt(5))/5;
b = (10 - 2*sqrt(5))/5;
A = (a^2,2*a);
B = (b^2,2*b);
O = (A + B)/2;
r = a + b;

for (y = -8; y <= 8; y = y + 0.2) {
  parab = parab--(y^2/4,y);
}

draw(parab,red);
draw((-2,0)--(16,0));
draw((0,-8)--(0,8));
draw(Circle(O,r));
draw(A--B);

dot("A", A, N);
dot("B", B, W);
[/asy]
Original LaTeX notation
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 $r.$

[asy]
unitsize(0.4 cm);

path parab = (16,-8);
real y;
pair A, B, O;
real a, b, r;

a = (10 + 2*sqrt(5))/5;
b = (10 - 2*sqrt(5))/5;
A = (a^2,2*a);
B = (b^2,2*b);
O = (A + B)/2;
r = a + b;

for (y = -8; y <= 8; y = y + 0.2) {
  parab = parab--(y^2/4,y);
}

draw(parab,red);
draw((-2,0)--(16,0));
draw((0,-8)--(0,8));
draw(Circle(O,r));
draw(A--B);

dot("$A$", A, N);
dot("$B$", B, W);
[/asy]

level

Level 5

type

Intermediate Algebra

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Source and history

task_id
78015003-8cbb-57f7-9bb3-8d80ff277300
task_key
intermediate~5falgebra--test--78015003-8cbb-57f7-9bb3-8d80ff277300
task_revision_id
1
config / split
intermediate_algebra / test

Official source

initial import