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MATH / Tangents are drawn from (1,3) to the parabola y²=4x, at A and B. Find the length AB. [asy] unitsize(0.4 cm); real upperparab (real x) return (sqrt(4*x)); real lowerparab (real x) r…

Problem

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

problem

Tangents are drawn from (1,3)(1,3) to the parabola y2=4x,y^2 = 4x, at AA and B.B. Find the length AB.AB. [asy] unitsize(0.4 cm); real upperparab (real x) { return (sqrt(4*x)); } real lowerparab (real x) { return (-sqrt(4*x)); } pair A, B, P; P = (1,3); A = ((7 + 3*sqrt(5))/2, upperparab((7 + 3*sqrt(5))/2)); B = ((7 - 3*sqrt(5))/2, upperparab((7 - 3*sqrt(5))/2)); draw(graph(upperparab,0,10)); draw(graph(lowerparab,0,10)); draw(interp(A,P,-0.8)--interp(A,P,1.2)); draw(interp(B,P,-1)--interp(B,P,1.5)); dot("AA", A, N); dot("BB", B, W); dot("(1,3)(1,3)", P, NW); [/asy]
Plain-text mathematical notation (without MathML)
Tangents are drawn from (1,3) to the parabola y²=4x, at A and B.  Find the length AB.

[asy]
unitsize(0.4 cm);

real upperparab (real x) {
  return (sqrt(4*x));
}

real lowerparab (real x) {
  return (-sqrt(4*x));
}

pair A, B, P;

P = (1,3);
A = ((7 + 3*sqrt(5))/2, upperparab((7 + 3*sqrt(5))/2));
B = ((7 - 3*sqrt(5))/2, upperparab((7 - 3*sqrt(5))/2));

draw(graph(upperparab,0,10));
draw(graph(lowerparab,0,10));
draw(interp(A,P,-0.8)--interp(A,P,1.2));
draw(interp(B,P,-1)--interp(B,P,1.5));

dot("A", A, N);
dot("B", B, W);
dot("(1,3)", P, NW);
[/asy]
Original LaTeX notation
Tangents are drawn from $(1,3)$ to the parabola $y^2 = 4x,$ at $A$ and $B.$  Find the length $AB.$

[asy]
unitsize(0.4 cm);

real upperparab (real x) {
  return (sqrt(4*x));
}

real lowerparab (real x) {
  return (-sqrt(4*x));
}

pair A, B, P;

P = (1,3);
A = ((7 + 3*sqrt(5))/2, upperparab((7 + 3*sqrt(5))/2));
B = ((7 - 3*sqrt(5))/2, upperparab((7 - 3*sqrt(5))/2));

draw(graph(upperparab,0,10));
draw(graph(lowerparab,0,10));
draw(interp(A,P,-0.8)--interp(A,P,1.2));
draw(interp(B,P,-1)--interp(B,P,1.5));

dot("$A$", A, N);
dot("$B$", B, W);
dot("$(1,3)$", P, NW);
[/asy]

level

Level 5

type

Intermediate Algebra

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Source and history

task_id
6e780eee-e32b-5658-8c23-c6eb5cf185af
task_key
intermediate~5falgebra--test--6e780eee-e32b-5658-8c23-c6eb5cf185af
task_revision_id
1
config / split
intermediate_algebra / test

Official source

initial import