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MATH / The parabola y=ax²+bx+c crosses the x-axis at (p,0) and (q,0), both to the right of the origin. A circle also passes through these two points. Let t be the length of the tangent fr…

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problem

The parabola y=ax2+bx+cy = ax^2 + bx + c crosses the xx-axis at (p,0)(p,0) and (q,0),(q,0), both to the right of the origin. A circle also passes through these two points. Let tt be the length of the tangent from the origin to the circle. Express t2t^2 in terms of one or more of the coefficients a,a, b,b, and c.c. [asy] unitsize(3 cm); pair A, O, T; real func (real x) { return ((x - 1)*(x - 2)); } A = (1.5,-0.4); O = (0,0); T = intersectionpoint(Circle(A,abs(A - (1,0))),arc(A/2,abs(A)/2,0,90)); draw(graph(func,0.5,2.5)); draw((-0.5,0)--(2.5,0)); draw((0,-1)--(0,1)); draw(Circle(A,abs(A - (1,0)))); draw(O--T); label("tt", T/3, N); dot(T); [/asy]
Plain-text mathematical notation (without MathML)
The parabola y=ax²+bx+c crosses the x-axis at (p,0) and (q,0), both to the right of the origin.  A circle also passes through these two points.  Let t be the length of the tangent from the origin to the circle.  Express t² in terms of one or more of the coefficients a, b, and c.

[asy]
unitsize(3 cm);

pair A, O, T;

real func (real x) {
  return ((x - 1)*(x - 2));
}

A = (1.5,-0.4);
O = (0,0);
T = intersectionpoint(Circle(A,abs(A - (1,0))),arc(A/2,abs(A)/2,0,90));

draw(graph(func,0.5,2.5));
draw((-0.5,0)--(2.5,0));
draw((0,-1)--(0,1));
draw(Circle(A,abs(A - (1,0))));
draw(O--T);

label("t", T/3, N);

dot(T);
[/asy]
Original LaTeX notation
The parabola $y = ax^2 + bx + c$ crosses the $x$-axis at $(p,0)$ and $(q,0),$ both to the right of the origin.  A circle also passes through these two points.  Let $t$ be the length of the tangent from the origin to the circle.  Express $t^2$ in terms of one or more of the coefficients $a,$ $b,$ and $c.$

[asy]
unitsize(3 cm);

pair A, O, T;

real func (real x) {
  return ((x - 1)*(x - 2));
}

A = (1.5,-0.4);
O = (0,0);
T = intersectionpoint(Circle(A,abs(A - (1,0))),arc(A/2,abs(A)/2,0,90));

draw(graph(func,0.5,2.5));
draw((-0.5,0)--(2.5,0));
draw((0,-1)--(0,1));
draw(Circle(A,abs(A - (1,0))));
draw(O--T);

label("$t$", T/3, N);

dot(T);
[/asy]

level

Level 5

type

Intermediate Algebra

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Source and history

task_id
39d82282-5e43-5ad0-afd1-1b5b366a3c6f
task_key
intermediate~5falgebra--test--39d82282-5e43-5ad0-afd1-1b5b366a3c6f
task_revision_id
1
config / split
intermediate_algebra / test

Official source

initial import