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2 years ago

The endpoints of a side of rectangle ABCD in the coordinate plane are at A (3,9) and

Mathematics

1 answer:

Montano1993 [528]2 years ago

5 0

Answer:

y = - 4x + 38

Step-by-step explanation:

A(3, 9)

B(5, 1)

$m_{AB}$ = $\frac{1-9}{5-3}$ = - 4

CD║AB

C(9, 2)

y - 2 = - 4(x - 9)

y = - 4x + 38

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Find the locus of a point such that the sum of its distance from the point ( 0 , 2 ) and ( 0 , -2 ) is 6.

jok3333 [9.3K]

Answer:

$\displaystyle \frac{x^2}{5}+\frac{y^2}{9}=1$

Step-by-step explanation:

We want to find the locus of a point such that the sum of the distance from any point P on the locus to (0, 2) and (0, -2) is 6.

First, we will need the distance formula, given by:

$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

Let the point on the locus be P(x, y).

So, the distance from P to (0, 2) will be:

$\begin{aligned} d_1&=\sqrt{(x-0)^2+(y-2)^2}\\\\ &=\sqrt{x^2+(y-2)^2}\end{aligned}$

And, the distance from P to (0, -2) will be:

$\displaystyle \begin{aligned} d_2&=\sqrt{(x-0)^2+(y-(-2))^2}\\\\ &=\sqrt{x^2+(y+2)^2}\end{aligned}$

So sum of the two distances must be 6. Therefore:

$d_1+d_2=6$

Now, by substitution:

$(\sqrt{x^2+(y-2)^2})+(\sqrt{x^2+(y+2)^2})=6$

Simplify. We can subtract the second term from the left:

$\sqrt{x^2+(y-2)^2}=6-\sqrt{x^2+(y+2)^2}$

Square both sides:

$(x^2+(y-2)^2)=36-12\sqrt{x^2+(y+2)^2}+(x^2+(y+2)^2)$

We can cancel the x² terms and continue squaring:

$y^2-4y+4=36-12\sqrt{x^2+(y+2)^2}+y^2+4y+4$

We can cancel the y² and 4 from both sides. We can also subtract 4y from both sides. This leaves us with:

$-8y=36-12\sqrt{x^2+(y+2)^2}$

We can divide both sides by -4:

$2y=-9+3\sqrt{x^2+(y+2)^2}$

Adding 9 to both sides yields:

$2y+9=3\sqrt{x^2+(y+2)^2}$

And, we will square both sides one final time.

$4y^2+36y+81=9(x^2+(y^2+4y+4))$

Distribute:

$4y^2+36y+81=9x^2+9y^2+36y+36$

The 36y will cancel. So:

$4y^2+81=9x^2+9y^2+36$

Subtracting 4y² and 36 from both sides yields:

$9x^2+5y^2=45$

And dividing both sides by 45 produces:

$\displaystyle \frac{x^2}{5}+\frac{y^2}{9}=1$

Therefore, the equation for the locus of a point such that the sum of its distance to (0, 2) and (0, -2) is 6 is given by a vertical ellipse with a major axis length of 3 and a minor axis length of √5, centered on the origin.

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3 years ago

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You watch a roulette wheel spin 270 consecutive times and the ball lands on a red slot each time. what is the probability that t

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18/37 if you're in Europe, 9/19 if you're in America, assuming the wheel isn't rigged. If the roulette wheel is a fair wheel, the odds on any given spin of the wheel is unaffected by any previous spins of the wheel. So no matter how many times the wheel has consecutively come up red, the next spin can still come up red, black, or green based upon the number of each color is present on the wheel. The difference between the odds for Europe vs America has to do with a quirk about roulette. In Europe, the roulette wheel has 37 slots, 18 red, 18 black, and 1 green. In America, the roulette wheel has 38 slots, 18 red, 18 black, and 2 green. Although in this case, I'd be wary about assuming that the wheel isn't rigged. The odds of 270 consecutive red on a fair wheel is only 1 in 3.09663x10^84 in Europe, or 1 in 4.14949x10^87 in America.

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kari74 [83]

I believe they are about the similar in shape unless you have answers to them?

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What is the equation of the line that passes through the point (-4,-2) and has a slope of -1/2?

Soloha48 [4]

Answer:

$y = - \frac{1}{2} x$