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MATH / The circle x²+y²=2 and the parabola y²=8x have two common tangents, forming four points of tangency. Find the area of the quadrilateral formed by the four points of tangency. [asy]…

Problem

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

problem

The circle x2+y2=2x^2 + y^2 = 2 and the parabola y2=8xy^2 = 8x have two common tangents, forming four points of tangency. Find the area of the quadrilateral formed by the four points of tangency. [asy] unitsize(0.8 cm); real upperparab (real x) { return (sqrt(8*x)); } real lowerparab (real x) { return (-sqrt(8*x)); } pair A, B, C, D; A = (-1,1); B = (2,4); C = (-1,-1); D = (2,-4); draw(graph(upperparab,0,3)); draw(graph(lowerparab,0,3)); draw(Circle((0,0),sqrt(2))); draw(interp(A,B,-0.2)--interp(A,B,1.2)); draw(interp(C,D,-0.2)--interp(C,D,1.2)); draw(A--C); draw(B--D); dot(A); dot(B); dot(C); dot(D); [/asy]
Plain-text mathematical notation (without MathML)
The circle x²+y²=2 and the parabola y²=8x have two common tangents, forming four points of tangency.  Find the area of the quadrilateral formed by the four points of tangency.

[asy]
unitsize(0.8 cm);

real upperparab (real x) {
  return (sqrt(8*x));
}

real lowerparab (real x) {
  return (-sqrt(8*x));
}

pair A, B, C, D;

A = (-1,1);
B = (2,4);
C = (-1,-1);
D = (2,-4);

draw(graph(upperparab,0,3));
draw(graph(lowerparab,0,3));
draw(Circle((0,0),sqrt(2)));
draw(interp(A,B,-0.2)--interp(A,B,1.2));
draw(interp(C,D,-0.2)--interp(C,D,1.2));
draw(A--C);
draw(B--D);

dot(A);
dot(B);
dot(C);
dot(D);
[/asy]
Original LaTeX notation
The circle $x^2 + y^2 = 2$ and the parabola $y^2 = 8x$ have two common tangents, forming four points of tangency.  Find the area of the quadrilateral formed by the four points of tangency.

[asy]
unitsize(0.8 cm);

real upperparab (real x) {
  return (sqrt(8*x));
}

real lowerparab (real x) {
  return (-sqrt(8*x));
}

pair A, B, C, D;

A = (-1,1);
B = (2,4);
C = (-1,-1);
D = (2,-4);

draw(graph(upperparab,0,3));
draw(graph(lowerparab,0,3));
draw(Circle((0,0),sqrt(2)));
draw(interp(A,B,-0.2)--interp(A,B,1.2));
draw(interp(C,D,-0.2)--interp(C,D,1.2));
draw(A--C);
draw(B--D);

dot(A);
dot(B);
dot(C);
dot(D);
[/asy]

level

Level 4

type

Intermediate Algebra

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Source and history

task_id
4b293ba3-f38c-5e09-87f7-d763a29b0746
task_key
intermediate~5falgebra--test--4b293ba3-f38c-5e09-87f7-d763a29b0746
task_revision_id
1
config / split
intermediate_algebra / test

Official source

initial import