let's notice something on this hyperbola, the fraction that is positive, is the fraction with the "y" variable, that simply means that the hyperbola is opening vertically, namely runs over the y-axis or it has a vertical traverse axis, which means, that, the foci will be a certain "c" distance from the center over the y-axis, well, with that mouthful, let's proceed.
![\bf \textit{hyperbolas, vertical traverse axis } \\\\ \cfrac{(y- k)^2}{ a^2}-\cfrac{(x- h)^2}{ b^2}=1 \qquad \begin{cases} center\ ( h, k)\\ vertices\ ( h, k\pm a)\\ c=\textit{distance from}\\ \qquad \textit{center to foci}\\ \qquad \sqrt{ a ^2 + b ^2}\\ asymptotes\quad y= k\pm \cfrac{a}{b}(x- h) \end{cases} \\\\[-0.35em] \rule{34em}{0.25pt}](https://tex.z-dn.net/?f=%5Cbf%20%5Ctextit%7Bhyperbolas%2C%20vertical%20traverse%20axis%20%7D%20%5C%5C%5C%5C%20%5Ccfrac%7B%28y-%20k%29%5E2%7D%7B%20a%5E2%7D-%5Ccfrac%7B%28x-%20h%29%5E2%7D%7B%20b%5E2%7D%3D1%20%5Cqquad%20%5Cbegin%7Bcases%7D%20center%5C%20%28%20h%2C%20k%29%5C%5C%20vertices%5C%20%28%20h%2C%20k%5Cpm%20a%29%5C%5C%20c%3D%5Ctextit%7Bdistance%20from%7D%5C%5C%20%5Cqquad%20%5Ctextit%7Bcenter%20to%20foci%7D%5C%5C%20%5Cqquad%20%5Csqrt%7B%20a%20%5E2%20%2B%20b%20%5E2%7D%5C%5C%20asymptotes%5Cquad%20y%3D%20k%5Cpm%20%5Ccfrac%7Ba%7D%7Bb%7D%28x-%20h%29%20%5Cend%7Bcases%7D%20%5C%5C%5C%5C%5B-0.35em%5D%20%5Crule%7B34em%7D%7B0.25pt%7D)
![\cfrac{(y-3)^2}{1}-\cfrac{(x+2)^2}{4}=1\implies \cfrac{[y-3]^2}{1^2}-\cfrac{[x-(-2)]^2}{2^2}=1~~ \begin{cases} h=-2\\ k=3\\ a=1\\ b=2 \end{cases} \\\\[-0.35em] ~\dotfill\\\\ c=\sqrt{a^2+b^2}\implies c=\sqrt{1+4}\implies c=\sqrt{5} \\\\\\ \stackrel{\textit{so then the foci are at}}{(-2~~,~~3\pm \sqrt{5})}\qquad \qquad \qquad \stackrel{\textit{and its vertices are at }}{(-2~~,~~3\pm 1)}\implies \begin{cases} (-2,4)\\ (-2,2) \end{cases}](https://tex.z-dn.net/?f=%5Ccfrac%7B%28y-3%29%5E2%7D%7B1%7D-%5Ccfrac%7B%28x%2B2%29%5E2%7D%7B4%7D%3D1%5Cimplies%20%5Ccfrac%7B%5By-3%5D%5E2%7D%7B1%5E2%7D-%5Ccfrac%7B%5Bx-%28-2%29%5D%5E2%7D%7B2%5E2%7D%3D1~~%20%5Cbegin%7Bcases%7D%20h%3D-2%5C%5C%20k%3D3%5C%5C%20a%3D1%5C%5C%20b%3D2%20%5Cend%7Bcases%7D%20%5C%5C%5C%5C%5B-0.35em%5D%20~%5Cdotfill%5C%5C%5C%5C%20c%3D%5Csqrt%7Ba%5E2%2Bb%5E2%7D%5Cimplies%20c%3D%5Csqrt%7B1%2B4%7D%5Cimplies%20c%3D%5Csqrt%7B5%7D%20%5C%5C%5C%5C%5C%5C%20%5Cstackrel%7B%5Ctextit%7Bso%20then%20the%20foci%20are%20at%7D%7D%7B%28-2~~%2C~~3%5Cpm%20%5Csqrt%7B5%7D%29%7D%5Cqquad%20%5Cqquad%20%5Cqquad%20%5Cstackrel%7B%5Ctextit%7Band%20its%20vertices%20are%20at%20%7D%7D%7B%28-2~~%2C~~3%5Cpm%201%29%7D%5Cimplies%20%5Cbegin%7Bcases%7D%20%28-2%2C4%29%5C%5C%20%28-2%2C2%29%20%5Cend%7Bcases%7D)
now let's check for the asymptotes.
![\bf y=3\pm \cfrac{1}{2}[x-(-2)]\implies y=3\pm \cfrac{1}{2}(x+2) \\\\[-0.35em] ~\dotfill\\\\ y=3+ \cfrac{1}{2}(x+2)\implies y=3+\cfrac{x+2}{2}\implies y=\cfrac{6+x+2}{2} \\\\\\ y=\cfrac{x+8}{2}\implies y=\cfrac{1}{2}x+4 \\\\[-0.35em] ~\dotfill\\\\ y=3- \cfrac{1}{2}(x+2)\implies y=3-\cfrac{(x+2)}{2}\implies y=\cfrac{6-(x+2)}{2} \\\\\\ y=\cfrac{6-x-2}{2}\implies y=\cfrac{-x+4}{2}\implies y=-\cfrac{1}{2}x+2](https://tex.z-dn.net/?f=%5Cbf%20y%3D3%5Cpm%20%5Ccfrac%7B1%7D%7B2%7D%5Bx-%28-2%29%5D%5Cimplies%20y%3D3%5Cpm%20%5Ccfrac%7B1%7D%7B2%7D%28x%2B2%29%20%5C%5C%5C%5C%5B-0.35em%5D%20~%5Cdotfill%5C%5C%5C%5C%20y%3D3%2B%20%5Ccfrac%7B1%7D%7B2%7D%28x%2B2%29%5Cimplies%20y%3D3%2B%5Ccfrac%7Bx%2B2%7D%7B2%7D%5Cimplies%20y%3D%5Ccfrac%7B6%2Bx%2B2%7D%7B2%7D%20%5C%5C%5C%5C%5C%5C%20y%3D%5Ccfrac%7Bx%2B8%7D%7B2%7D%5Cimplies%20y%3D%5Ccfrac%7B1%7D%7B2%7Dx%2B4%20%5C%5C%5C%5C%5B-0.35em%5D%20~%5Cdotfill%5C%5C%5C%5C%20y%3D3-%20%5Ccfrac%7B1%7D%7B2%7D%28x%2B2%29%5Cimplies%20y%3D3-%5Ccfrac%7B%28x%2B2%29%7D%7B2%7D%5Cimplies%20y%3D%5Ccfrac%7B6-%28x%2B2%29%7D%7B2%7D%20%5C%5C%5C%5C%5C%5C%20y%3D%5Ccfrac%7B6-x-2%7D%7B2%7D%5Cimplies%20y%3D%5Ccfrac%7B-x%2B4%7D%7B2%7D%5Cimplies%20y%3D-%5Ccfrac%7B1%7D%7B2%7Dx%2B2)
Answer:
D∪∪
Step-by-step explanation:
x²+x-72=0
x²+9x-8x-72=0
x(x+9)-8(x+9)=0
(x+9)(x-8)=0
x=-9,8
Answer:
True
Step-by-step explanation:
A matrix is a rectangular array in which elements are arranged in rows and columns.
An augmented matrix is a matrix in which same row operations are performed on both the sides of equal signs in the given linear system of equations.
Elementary row operations are the operations like multiplication or division which are performed in the original matrix to get the elementary matrix.
True, elementary row operations on an augmented matrix never change the solution set of the associated linear system as the elementary row operations replace a linear system with an equivalent linear system.
