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MATH / Let T=(a,9−a²) be a point on the parabola y=9−x², for some positive rational number a. [asy] unitsize(1 cm); real parab (real x) return(4 - x^2); pair T, X, Y; real m, t; t = 0.8; …

Problem

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problem

Let T=(a,9−a2)T = (a, 9 - a^2) be a point on the parabola y=9−x2,y = 9 - x^2, for some positive rational number a.a. [asy] unitsize(1 cm); real parab (real x) { return(4 - x^2); } pair T, X, Y; real m, t; t = 0.8; m = -2*t; T = (t,parab(t)); X = extension(T, T + (1,m), (0,0), (1,0)); Y = extension(T, T + (1,m), (0,0), (0,1)); draw(graph(parab,0,2)); draw(X--Y); draw((0,0)--(4,0)); draw((0,0)--(0,5)); dot("T=(a,9−a2)T = (a, 9 - a^2)", T, NE); [/asy] The tangent to the parabola at TT and the coordinate axes form a triangle with area 25. Find a.a.
Plain-text mathematical notation (without MathML)
Let T=(a,9−a²) be a point on the parabola y=9−x², for some positive rational number a.

[asy]
unitsize(1 cm);

real parab (real x) {
  return(4 - x^2);
}

pair T, X, Y;
real m, t;

t = 0.8;
m = -2*t;
T = (t,parab(t));
X = extension(T, T + (1,m), (0,0), (1,0));
Y = extension(T, T + (1,m), (0,0), (0,1));

draw(graph(parab,0,2));
draw(X--Y);
draw((0,0)--(4,0));
draw((0,0)--(0,5));

dot("T=(a,9−a²)", T, NE);
[/asy]

The tangent to the parabola at T and the coordinate axes form a triangle with area 25.  Find a.
Original LaTeX notation
Let $T = (a, 9 - a^2)$ be a point on the parabola $y = 9 - x^2,$ for some positive rational number $a.$

[asy]
unitsize(1 cm);

real parab (real x) {
  return(4 - x^2);
}

pair T, X, Y;
real m, t;

t = 0.8;
m = -2*t;
T = (t,parab(t));
X = extension(T, T + (1,m), (0,0), (1,0));
Y = extension(T, T + (1,m), (0,0), (0,1));

draw(graph(parab,0,2));
draw(X--Y);
draw((0,0)--(4,0));
draw((0,0)--(0,5));

dot("$T = (a, 9 - a^2)$", T, NE);
[/asy]

The tangent to the parabola at $T$ and the coordinate axes form a triangle with area 25.  Find $a.$

level

Level 4

type

Intermediate Algebra

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Source and history

task_id
41175514-9237-5966-8d8b-26f39b1ed378
task_key
intermediate~5falgebra--test--41175514-9237-5966-8d8b-26f39b1ed378
task_revision_id
1
config / split
intermediate_algebra / test

Official source

initial import