Answer:
2a=1
Step-by-step explanation:
The constant difference for a hyperbola
is ![2a.](https://tex.z-dn.net/?f=2a.)
1. Since hyperbola has foci (-3.5, 0) and (3.5, 0), then
Note that
then
![\sqrt{a^2+b^2}=3.5\Rightarrow a^2+b^2=3.5^2.](https://tex.z-dn.net/?f=%5Csqrt%7Ba%5E2%2Bb%5E2%7D%3D3.5%5CRightarrow%20a%5E2%2Bb%5E2%3D3.5%5E2.)
2. Since point (3.5,24) lies on the hyperbola, then
![\dfrac{3.5^2}{a^2}-\dfrac{24^2}{b^2}=1.](https://tex.z-dn.net/?f=%5Cdfrac%7B3.5%5E2%7D%7Ba%5E2%7D-%5Cdfrac%7B24%5E2%7D%7Bb%5E2%7D%3D1.)
3. Solve the system of two equations:
![\left\{\begin{array}{l}a^2+b^2=3.5^2\\\dfrac{3.5^2}{a^2}-\dfrac{24^2}{b^2}=1\end{array}\right.](https://tex.z-dn.net/?f=%5Cleft%5C%7B%5Cbegin%7Barray%7D%7Bl%7Da%5E2%2Bb%5E2%3D3.5%5E2%5C%5C%5Cdfrac%7B3.5%5E2%7D%7Ba%5E2%7D-%5Cdfrac%7B24%5E2%7D%7Bb%5E2%7D%3D1%5Cend%7Barray%7D%5Cright.)
From the 1st equation,
![b^2=3.5^2-a^2,](https://tex.z-dn.net/?f=b%5E2%3D3.5%5E2-a%5E2%2C)
then
![\dfrac{3.5^2}{a^2}-\dfrac{24^2}{3.5^2-a^2}=1,\\ \\\dfrac{3.5^2(3.5^2-a^2)-24^2a^2}{a^2(3.5^2-a^2)}=1,\\ \\3.5^4-3.5^2a^2-24^2a^2=3.5^2a^2-a^4,\\ \\a^4-a^2(2\cdot 3.5^2+24^2)+3.5^4=0,\\ \\a^4-600.5a^2+150.0625=0,\\ \\D=(-600.5)^2-4\cdot 150.0625=360000,\\ \\a^2_{1,2}=\dfrac{600.5\pm 600}{2}=\dfrac{1}{4},600\dfrac{1}{4}.](https://tex.z-dn.net/?f=%5Cdfrac%7B3.5%5E2%7D%7Ba%5E2%7D-%5Cdfrac%7B24%5E2%7D%7B3.5%5E2-a%5E2%7D%3D1%2C%5C%5C%20%5C%5C%5Cdfrac%7B3.5%5E2%283.5%5E2-a%5E2%29-24%5E2a%5E2%7D%7Ba%5E2%283.5%5E2-a%5E2%29%7D%3D1%2C%5C%5C%20%5C%5C3.5%5E4-3.5%5E2a%5E2-24%5E2a%5E2%3D3.5%5E2a%5E2-a%5E4%2C%5C%5C%20%5C%5Ca%5E4-a%5E2%282%5Ccdot%203.5%5E2%2B24%5E2%29%2B3.5%5E4%3D0%2C%5C%5C%20%5C%5Ca%5E4-600.5a%5E2%2B150.0625%3D0%2C%5C%5C%20%5C%5CD%3D%28-600.5%29%5E2-4%5Ccdot%20150.0625%3D360000%2C%5C%5C%20%5C%5Ca%5E2_%7B1%2C2%7D%3D%5Cdfrac%7B600.5%5Cpm%20600%7D%7B2%7D%3D%5Cdfrac%7B1%7D%7B4%7D%2C600%5Cdfrac%7B1%7D%7B4%7D.)
For ![a^2=\dfrac{1}{4},\ b^2=3.5^2-0.25=12.](https://tex.z-dn.net/?f=a%5E2%3D%5Cdfrac%7B1%7D%7B4%7D%2C%5C%20b%5E2%3D3.5%5E2-0.25%3D12.)
For
this is impossible, then
is extra solution.
Hence,
and ![2a=1.](https://tex.z-dn.net/?f=2a%3D1.)
Answer:
g(x) = |x| -9
Step-by-step explanation:
In the form ...
g(x) = a|x -h| +k
the parameters are ...
- a: the vertical scale factor (1)
- h: the right shift of the vertex (0)
- k: the upward shift of the vertex (-9)
The vertex of |x| is (0, 0). The vertex of g(x) is (0, -9), so there has been a vertical shift of -9 and no horizontal shift. The slopes of the lines on the graph of g(x) have a magnitude of 1, so the vertical scale is not changed from the original function (a=1).
Then the translated function is ...
g(x) = |x| -9
The answer is 31/35
Hope it helps!
Answer:
![-30](https://tex.z-dn.net/?f=-30)
Step-by-step explanation:
![-3(-4)(-2.5)](https://tex.z-dn.net/?f=-3%28-4%29%28-2.5%29)
Apply rule ![-(-a) = a](https://tex.z-dn.net/?f=-%28-a%29%20%3D%20a)
![=-3*4*2.5](https://tex.z-dn.net/?f=%3D-3%2A4%2A2.5)
Multiply the numbers: ![3*4*2.5=30](https://tex.z-dn.net/?f=3%2A4%2A2.5%3D30)
![=-30](https://tex.z-dn.net/?f=%3D-30)
Hello : let A(3,0) B(0,-5)<span>
the slope is : (YB - YA)/(XB -XA)
(-5-0)/(0-3) = 5/3
</span>
<span> an equation, in the slope intercept form is : y - (-5) = (5/3)(x-0)
y = (5/3)x - 5</span>