SAT Question - "In the xy-coordinate plane..."

[Guest]

Hi,

I'm stumped on this question:

In the xy-coordinate plane, the graph of x = y^2 - 4 intersects line L at (0, p) and (5, t). What is the greatest possible value of the slope of L?

I solve for y by plugging in the x values. I now have two points: (0,2) and (5,3). The slope is 1/5. From there I don't know how to find out the greatest possible slope for L.

The answer is 1, I just don't know how to arrive there.

Thanks for your help!

- Styles

 

[stapel]

Note: This is a sideways parabola!

Graph it. You'll find that, for x = 0, there are two solutions, as for when x = 5. These four points give you four line segments. You need to find the largest slope of the four.

Eliz.

 

[Guest]

I think I understand what you're saying.

Instead of only two points (0,2) and (5,3) I also have (0,-2) and (5,-3). The slope of (0,-2) and (5,3) is the largest, 1.

Thanks for your help!

 

[soroban]

Hello, Styles!

This is a strange question.
The way it is worded, there are only a few possible slopes for line L.
. . We just have to select the largest one.

In the xy-coordinate plane, the graph of $\displaystyle x\:=\:y^2\,-\,4$ intersects line $\displaystyle L$ at $\displaystyle A(0,\,p)$ and $\displaystyle B(5,\,t)$.
What is the greatest possible value of the slope of $\displaystyle L$?

Make a sketch.
We find that $\displaystyle p\,=\,\pm2$ and $\displaystyle t\,=\,\pm3$

              |         B /
              |         *(5,3)
             A*       /
         *    |(0,2) /
       *      |     /
              |    /
    --*-------+---/-----+--
      4       |  /      5
       *      | /
         *    |/
            A'*
            / |(0,-2)   *(5,-3)
              |         B'

There are two choices for point $\displaystyle (0,p):\;A(0,2)$ or $\displaystyle A'(0.-2)$
There are two choices for point $\displaystyle (5,t):\;B(5,3)$ or $\displaystyle B'(5,-3)$

There are <u>four</u> possible choices for line $\displaystyle L$.
. . Line $\displaystyle A'B'$ has a negative slope.
. . Line $\displaystyle AB'$ has a very negative slope.
. . Line $\displaystyle AB$ has a positive slope.
. . But line $\displaystyle A'B$ has a very positive slope . . . That's the one! .Find its slope.