1985 AMC 12 Problem 19

Question

Consider the graphs of

y=Ax^2

and

y^2+3=x^2+4y,

where

A

is a positive constant and

x

and

y

are real variables. In how many points do the two graphs intersect?

exactly

4

exactly

2

at least

1,

but the number varies for different positive values of

A

0

for at least one positive value of

A

none of these

Use

x^2=\frac{y}{A}

to eliminate

x^2

Show that the resulting quadratic has two distinct positive

y

-roots

Substituting

x^2=\frac{y}{A}

into the second equation gives

Ay^2-(4A+1)y+3A=0.

Its discriminant is

4A^2+8A+1\gt0.

Its roots have positive product

3

and positive sum

\frac{4A+1}{A},

so both roots are positive and distinct. For each root

y,

the equation

x^2=\frac{y}{A}

gives two distinct values of

x.

Thus there are exactly four intersection points.

Therefore the correct answer is A.

Problem 19 in other years