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In the xy-coordinate plane, the graph of y = -x^2 + 9 inters  [#permalink]

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In the xy-coordinate plane, the graph of y = -x^2 + 9 intersects line L at (p,5) and (t,-7). What is the least possible value of the slope of line L?

A. 6
B. 2
C. -2
D. -6
E. -10

Originally posted by usre123 on 17 Apr 2013, 01:40.
Last edited by Bunuel on 17 Apr 2013, 01:46, edited 2 times in total.
Edited the question.
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Re: In the xy-coordinate plane, the graph of y = -x^2 + 9 inters  [#permalink]

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usre123 wrote:
In the xy-coordinate plane, the graph of y = -x^2 + 9 intersects line L at (p,5) and (t,-7). What is the least possible value of the slope of line L?

A. 6
B. 2
C. -2
D. -6
E. -10

We need to find out the value of p and L to get to the slope.

Line L and Graph y intersect at point (p,5). hence, x= p and Y=5 should sactisfy the graph. soliving

5 = -p2 +9

p2 = 4
p = + or - 2

simillarly point (t,-7) should satisfy the equation. hence x=t and Y=-7.

-7 = -t2+9
t = +or - 4
considering p = -2 and t =4, the least slope is (-7-5)/(4-2) = -6

IMO option D is correct answer.
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Re: In the xy-coordinate plane, the graph of y = -x^2 + 9 inters  [#permalink]

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A classic geometry question and a good one too.

Let's solve this First of all, we know that the graph represented by the equation (E) $$y = -x^2 + 9$$ intersects line L in two points : A (p,5) and B (t, -7).

This automatically tells us that points A and B satisfy equation (E).

As such, we'll have the following two equations :

(1) : $$5 = - p^2 + 9$$ which yields $$p^2 = 4$$ meaning that $$p = 2$$ or $$p = -2$$
(2) : $$- 7 = -t^2 + 9$$ which yields $$t^2 = 16$$ meaning that $$t = 4$$ or $$t = -4$$

(Remember that even powers hide the sign of the base so always include the negative answers unless told otherwise by the question stem)

Considering the answers we've obtained, we have 4 possibilities for points A and B :

Possibility n°1 : A (2,5) and B (4,-7)
Possibility n°2 : A (2,5) and B (-4, -7)
Possibility n°3 : A (-2,5) and B (4, -7)
Possibility n°4 : A (-2,5) and B (-4,-7)

As a reminder :

1/ In a xy-coordinate plane, a line is defined by the following equation : $$y = a*x + b$$ (with a being the slope of the line)
2/ Given two points $$A (x1,y1)$$ and $$B (x2,y2)$$, the slope of a line can be computed as such :$$a = \frac{(y2-y1)}{(x2-x1)}$$

So considering reminder n°2, for each of the possibilities above we get as a value for the slope of line L :

Possibility n°1 : a = -6
Possibility n°2 : a = 2
Possibility n°3 : a = -2
Possibility n°4 : a = 6

As such the least possible value of the slope of line L is -6, which is answer choice D.

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Re: In the xy-coordinate plane, the graph of y = -x^2 + 9 inters  [#permalink]

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$$y = -x^2 + 9$$ intersects line $$y=5$$ in two points (2,5) and (-2,5)
intersects line $$y=-7$$ in two points (4,7) and (-4,7)

As you can see in the imgage, we have to pick 2 numbers that minimize the slope. The slope must be negative and as "vertical" as possible => points ($$2,5) (4,-7)$$
Slope $$s=\frac{y1-y2}{x1-x2}=\frac{5+7}{2-4}=-\frac{12}{6}=-6$$
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Re: In the xy-coordinate plane, the graph of y = -x^2 + 9 inters  [#permalink]

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_________________ Re: In the xy-coordinate plane, the graph of y = -x^2 + 9 inters   [#permalink] 05 Jan 2019, 15:56
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