SOLUTION: A Parabola with the equation {{{y=ax^2}}} passes through three vertices of a square. if the area of the square = 18, find the value of {{{a}}} in {{{y=ax^2}}}.

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: A Parabola with the equation {{{y=ax^2}}} passes through three vertices of a square. if the area of the square = 18, find the value of {{{a}}} in {{{y=ax^2}}}.      Log On


   



Question 173551: A Parabola with the equation y=ax%5E2 passes through three vertices of a square. if the area of the square = 18, find the value of a in y=ax%5E2.
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
A Parabola with the equation y=ax%5E2 passes through three vertices of a square. if the area of the square = 18, find the value of a in y=ax%5E2.

The formula for the area of a square, where s 
is the length of each side is 

A+=+s%5E2

Substitute 18 for A

18+=+s%5E2

Take square roots of both sides

sqrt%2818%29=sqrt%28s%5E2%29

sqrt%2818%29=s

So each side is sqrt%2818%29

We draw the square:



Let's draw a diagonal:



Use the Pythagorean theorem to find the length 
of the diagonal:

diagonal%5E2=+%28sqrt%2818%29%29%5E2+%2B+%28sqrt%2818%29%29%5E2
diagonal%5E2=+18%2B18
diagonal%5E2=36
Taking square roots:
diagonal=6

Now since the diagonal is 6, let's tilt the square
45° and put the diagonal along the y-axis with a 
corner at the origin, like this: 



We draw the other diagonal:



Since the vertical diagonal is 6 units,
so is the horizontal diagonal, so
the two vertices at the end of the
horizontal diagonal are matrix%281%2C1%2C%22%283%2C3%29%22%29
and matrix%281%2C1%2C%22%28-3%2C3%29%22%29 

We put the coordinates at these corners:



Now we can roughly sketch in the parabola
we are looking for:

We put the coordinates at these corners:



Since the parabola has the form y=ax%5E2,
and goes through matrix%281%2C1%2C%22%283%2C3%29%22%29, 
we substitute x=3 and y=3

y=ax%5E2
3=a%283%29%5E2
3=a%289%29
3=9a
3%2F9=a
1%2F3=a  

The value of a is therefore 1%2F3 and
the equation of the parabola is

y=1%2F3x%5E2  

Edwin