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.