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3 years ago

The coordinates of 4 points are A(0,9), B(k + 1, k + 4), C(2k, k + 3) and

Mathematics

1 answer:

WITCHER [35]3 years ago

4 0

9514 1404 393

Answer:

(a) 2 or 3

(b) -13

Step-by-step explanation:

(a) A, B, and C are collinear if there is some constant 'c' such that ...

B - A = c(C - A)

This will give rise to two equations in 'c' and 'k'.

(k +1, k +4) -(0, 9) = c((2k, k+3) -(0, 9))

(k +1, k -5) = c(2k, k -6)

And the two equations are ...

k +1 = 2ck

k -5 = c(k -6)

From the first, we find ...

c = (k +1)/(2k)

From the second, we find ...

c = (k -5)/(k -6)

Equating these expressions for c gives ...

(k +1)/(2k) = (k -5)/(k -6)

(k +1)(k -6) = 2k(k -5) . . . . . . . multiply by 2k(k-6)

k^2 -5k -6 = 2k^2 -10k

0 = k^2 -5k +6 = (k -3)(k -2)

Solutions that make the factors zero are ...

k = 3 or k = 2

There are two values of k that make the points collinear: 2 and 3.

(b) AB is parallel to CD when the slopes of the lines between them are the same

slope of AB = (k+4 -9)/(k+1 -0) = (k -5)/(k +1)

slope of CD = ((k +6 -(k +3))/(2k +2 -2k) = 3/2

Equating these slopes and solving for k, we get ...

(k -5)/(k +1) = 3/2

2(k -5) = 3(k +1)

2k -10 = 3k +3

-13 = k

AB is parallel to CD when k = -13.

_____

The attached shows the collinear points on the red and blue lines. The points that make parallel lines are shown on the purple lines. (The purple circles correspond to points D for k=2 and 3, so are irrelevant.) Point D is not labeled with its coordinates, (-24, -7).

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Mr Thompson please help with my new HW

sesenic [268]

Answers:

C) x = plus/minus 11
B) No real solutions
C) Two solutions
A) One solution
The value <u> 18 </u> goes in the first blank. The value <u> 17 </u> goes in the second blank.

========================================================

Explanations:

Note how (11)^2 = (11)*(11) = 121 and also (-11)^2 = (-11)*(-11) = 121. The two negatives multiply to a positive. So that's why the solution is x = plus/minus 11. The plus minus breaks down into the two equations x = 11 or x = -11.
There are no real solutions here because the left hand side can never be negative, no matter what real number you pick for x. As mentioned in problem 1, squaring -11 leads to a positive number 121. The same idea applies here as well.
The two solutions are x = 0 and x = -2. We set each factor equal to zero through the zero product property. Then we solve each equation for x. The x+2 = 0 leads to x = -2.
We use the zero product property here as well. We have a repeated factor, so we're only solving one equation and that is x-3 = 0 which leads to x = 3. The only root is x = 3.
Apply the FOIL rule on (x+1)(x+17) to end up with x^2+17x+1x+17 which simplifies fully to x^2+18x+17. The middle x coefficient is 18, while the constant term is 17.

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Step-by-step explanation:

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Answer:

Length (parallel to the x-axis): $2 \sqrt{2}$ ;

Height (parallel to the y-axis): $4\sqrt{2}$ .

Step-by-step explanation:

Let the top-right vertice of this rectangle $(x,y)$ . $x, y >0$ . The opposite vertice will be at $(-x, -y)$ . The length the rectangle will be $2x$ while its height will be $2y$ .

Function that needs to be maximized: $f(x, y) = (2x)(2y) = 4xy$ .

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(Note that this method still applies even if there are more than two variables.)

That's two equations for three variables. Don't panic. The constraint itself acts as the third equation of this system:

$g(x, y) = k$ .

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If $x = 0$ , the area of the rectangle will equal to zero. That's likely not a solution.

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