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MATH / The parametric curve (x,y)=(sint,sin2t) is plotted below.

Problem

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

problem

The parametric curve (x,y)=(sint,sin2t)(x,y) = (\sin t, \sin 2t) is plotted below. [asy] unitsize(2 cm); pair moo (real t) { return (sin(t),sin(2*t)); } real t; path foo = moo(0); for (t = 0; t <= 2*pi + 0.1; t = t + 0.1) { foo = foo--moo(t); } draw((-1.2,0)--(1.2,0)); draw((0,-1.2)--(0,1.2)); draw(foo,red); [/asy] This curve can also be expressed in the form ax4−bx2+cy2=0,ax^4 - bx^2 + cy^2 = 0,where a,a, b,b, and cc are positive integers. Find the smallest possible value of a+b+c.a + b + c.
Plain-text mathematical notation (without MathML)
The parametric curve (x,y)=(sint,sin2t) is plotted below.

[asy]
unitsize(2 cm);

pair moo (real t) {
  return (sin(t),sin(2*t));
}

real t;
path foo = moo(0);

for (t = 0; t <= 2*pi + 0.1; t = t + 0.1) {
  foo = foo--moo(t);
}

draw((-1.2,0)--(1.2,0));
draw((0,-1.2)--(0,1.2));
draw(foo,red);
[/asy]

This curve can also be expressed in the form
ax⁴−bx²+cy²=0,where a, b, and c are positive integers.  Find the smallest possible value of a+b+c.
Original LaTeX notation
The parametric curve $(x,y) = (\sin t, \sin 2t)$ is plotted below.

[asy]
unitsize(2 cm);

pair moo (real t) {
  return (sin(t),sin(2*t));
}

real t;
path foo = moo(0);

for (t = 0; t <= 2*pi + 0.1; t = t + 0.1) {
  foo = foo--moo(t);
}

draw((-1.2,0)--(1.2,0));
draw((0,-1.2)--(0,1.2));
draw(foo,red);
[/asy]

This curve can also be expressed in the form
\[ax^4 - bx^2 + cy^2 = 0,\]where $a,$ $b,$ and $c$ are positive integers.  Find the smallest possible value of $a + b + c.$

level

Level 5

type

Precalculus

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Source and history

task_id
9e980e35-fc46-5d89-bb9c-faf5926a14aa
task_key
precalculus--test--9e980e35-fc46-5d89-bb9c-faf5926a14aa
task_revision_id
1
config / split
precalculus / test

Official source

initial import