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Serggg [28]
3 years ago
11

Two parabolas have the same focus, namely the point $(3,-28).$ Their directrices are the $x$-axis and the $y$-axis, respectively

. Compute the slope of their common chord.

Mathematics
1 answer:
jeyben [28]3 years ago
3 0

Answer:

the slope of the common cord is: -1

Step-by-step explanation:

Given the focus (a,b) = (3,-28)

  • for a parabola with directrix at x-axis the equation will be

\left(x-a\right)^{2}+\left(y-b\right)^{2}=y^{2}

\left(x-a\right)^{2}+\left(y-b\right)^{2}-y^{2}=0

  • for a parabola with directrix at y-axis the equation will be

\left(x-a\right)^{2}+\left(y-b\right)^{2}=x^{2}

\left(x-a\right)^{2}+\left(y-b\right)^{2}-x^{2}=0

The common chord is the line between two points where the two parabolas intersect. For intersection, we can equate the two parabolas!

In other words, at the point of intersection of these two parabolas the values of the two parabolas will be the same.

\left(x-a\right)^{2}+\left(y-b\right)^{2}-y^{2}=\left(x-a\right)^{2}+\left(y-b\right)^{2}-x^{2}

\left(x-3\right)^{2}+\left(y+28\right)^{2}-y^{2}=\left(x-3\right)^{2}+\left(y+28\right)^{2}-x^{2}

we can now simplify the equation. (we can see that (x-3) and (y+28) both cancel out by -(x-3) and -(y+28))

-y^{2}=-x^{2}

y=\sqrt{x^{2}}

y=\pm x

this the equation of the common cord. but we need to select whether its

y=+ x or y=-x.

This can be found by realizing that the focus lies on the 4th quadrant of the xy-plane! And the equation y=-x also generates a line that exists in the 2nd and 4th quadrant.

Hence the slope of the common cord is the slope of the line y=-x

that is : -1

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3 0
3 years ago
Read 2 more answers
IM BEGGING YOU PLEASE HELP PLEASE HELP I NEED HELP PLEASE
I am Lyosha [343]

Answer:

area of shaded region: 21.07 ft²

<u>Given for semi circle:</u>

  • radius of semi-circle: 7 ft

<u>Given for rectangle:</u>

  • width: 7 ft
  • length: 14 ft

<u>Find area of rectangle:</u>

Length * Width

14 * 7

98 ft²

<u>Find the area of semi circle:</u>

1/2 * π * r²

1/2 * 3.14 * (7)²

76.93 ft²

<u>Area of shaded region:</u>

area of rectangle - area of semi-circle

98 ft² - 76.93 ft²

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8 0
2 years ago
How about identify witch stamt is true A. -2.6&gt;2.521 B.-0.4&gt;-0.004 C.-8.0329&gt;-8.045 D.-36.5&gt;-36.07
astra-53 [7]
<span>A. -2.6 > 2.521 false
B.-0.4 > -0.004  false
C.-8.0329 > -8.045 true
D.-36.5 > -36.07 false

answer

</span>C.-8.0329 > -8.045 
5 0
3 years ago
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