1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Maslowich
1 year ago
13

Find the center that eliminates the linear terms in the translation of 4x^2 - y^2 + 24x + 4y + 28 = 0.(-3, 2)(-3,- 2)(4, 0)

Mathematics
1 answer:
baherus [9]1 year ago
4 0

Step 1

Given;

4x^2-y^2+24x+4y+28=0

Required; To find the center that eliminates the linear terms

Step 2

\begin{gathered} 4x^2-y^2+24x+4y=-28 \\ 4x^2+24x-y^2+4y=-28 \\ Complete\text{ the square }; \\ 4x^2+24x \\ \text{use the form ax}^2+bx\text{ +c} \\ \text{where} \\ a=4 \\ b=24 \\ c=0 \end{gathered}\begin{gathered} consider\text{ the vertex }form\text{ of a }parabola \\ a(x+d)^2+e \\ d=\frac{b}{2a} \\ d=\frac{24}{2\times4} \\ d=\frac{24}{8} \\ d=3 \end{gathered}\begin{gathered} Find\text{ the value of e using }e=c-\frac{b^2}{4a} \\ e=0-\frac{24^2}{4\times4} \\ e=0-\frac{576}{16}=-36 \end{gathered}

Step 3

Substitute a,d,e into the vertex form

\begin{gathered} a(x+d)^2+e \\ 4(x+_{}3)^2-36 \end{gathered}\begin{gathered} 4(x+3)^2-36-y^2+4y=-28 \\ 4(x+3)^2-y^2+4y=\text{ -28+36} \\  \\  \end{gathered}

Step 4

Completing the square for -y²+4y

\begin{gathered} \text{use the form ax}^2+bx\text{ +c} \\ \text{where} \\ a=-1 \\ b=4 \\ c=0 \end{gathered}\begin{gathered} consider\text{ the vertex }form\text{ of a }parabola \\ a(x+d)^2+e \\ d=\frac{b}{2a} \\ d=\text{ }\frac{4}{2\times-1} \\ d=\frac{4}{-2} \\ d=-2 \end{gathered}\begin{gathered} Find\text{ the value of e using }e=c-\frac{b^2}{4a} \\ e=0-\frac{4^2}{4\times(-1)} \\  \\ e=0-\frac{16}{-4} \\ e=4 \end{gathered}

Step 5

Substitute a,d,e into the vertex form

\begin{gathered} a(y+d)^2+e \\ =-1(y+(-2))^2+4 \\ =-(y-2)^2+4 \end{gathered}

Step 6

\begin{gathered} 4(x+3)^2-y^2+4y=\text{ -28+36} \\ 4(x+3)^2-(y-2)^2+4=-28+36 \\ 4(x+3)^2-(y-2)^2=-28+36-4 \\ 4(x+3)^2-(y-2)^2=4 \\ \frac{4(x+3)^2}{4}-\frac{(y-2)^2}{4}=\frac{4}{4} \\ (x+3)^2-\frac{(y-2)^2}{2^2}=1 \end{gathered}

Step 7

\begin{gathered} \frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1 \\ \text{This is the }form\text{ of a hyperbola.} \\ \text{From here } \\ a=1 \\ b=2 \\ k=2 \\ h=-3 \end{gathered}

Hence the answer is (-3,2)

You might be interested in
What is the value of x?
hodyreva [135]
Where is the equation
4 0
3 years ago
Solve the inequality 2x < 8 - 4(2 - x)
nevsk [136]
Remember you can' do anything to boht sides, but when times negative, flip sign
distributive property
a(b+c)=ab+ac

distribute the -4
-8+4x

now we have
2x<8-8+4x
2x<4x
minus 2x both sides
0<2x
therefor
0<x

answer is x>0
5 0
3 years ago
Choose the function whose graph is given below.
torisob [31]

Answer:

A y=tanx

Step-by-step explanation:

mark me brainliest pl

7 0
2 years ago
Find the area of the shaded region ​
o-na [289]

so hmmm let's get the area of the whole hexagon, and then get the area of the circle inside it, then <u>subtract the area of the circle from that of the hexagon's</u>, what's leftover is what we didn't subtract, namely the shaded part.

\textit{area of a regular polygon}\\\\ A=\cfrac{1}{4}ns^2\cot\stackrel{\stackrel{degrees}{\downarrow }}{\left( \frac{180}{n} \right)}~ \begin{cases} n=\textit{number of sides}\\ s=\textit{length of a side}\\[-0.5em] \hrulefill\\ n=\stackrel{hexagon}{6}\\ s=\frac{9}{2} \end{cases}\implies A=\cfrac{1}{4}(6)\left( \cfrac{9}{2} \right)^2 \cot\left( \cfrac{180}{6} \right)

A=\cfrac{1}{4}(6)\cfrac{9^2}{2^2} \cot(30^o)\implies A=\cfrac{243}{8}\cot(30^o)\implies A=\cfrac{243\sqrt{3}}{8} \\\\[-0.35em] ~\dotfill\\\\ \textit{area of circle}\\\\ A=\pi r^2~~ \begin{cases} r=radius\\[-0.5em] \hrulefill\\ r=\frac{4}{5} \end{cases}\implies A=\pi \left( \cfrac{4}{5} \right)^2\implies A=\cfrac{16\pi }{25} \\\\[-0.35em] ~\dotfill

\stackrel{\textit{area of the hexagon}}{\cfrac{243\sqrt{3}}{8}}~~ - ~~\stackrel{\textit{area of the circle}}{\cfrac{16\pi }{25}}\implies \cfrac{6075\sqrt{3}-128\pi }{200}

5 0
2 years ago
6m+10p when m=8 and p=14
Luba_88 [7]

Idk sorryyyyyyyyyyyyyyyyyyy
3 0
3 years ago
Read 2 more answers
Other questions:
  • What is another way of asking, "What is the log base of sixty four?"
    9·1 answer
  • Which equations have a value less than 6,766? A. one fourth x 6,766 = ________ B. 6 x 6,766 = ________ C. one half x 6,766 = ___
    14·1 answer
  • Y = (1/4)x^2 - (1/2)lnx..over the interval (1, 7e) ...what is the arc length ?
    6·1 answer
  • What is the median of 45,50,47,52,53,45,51
    5·2 answers
  • What is fifteen percent of seventy dollars
    9·2 answers
  • Need help in all asap plz
    6·1 answer
  • Help please. Thanks!
    10·2 answers
  • What are vertical angles always equal to
    13·1 answer
  • Please help me! I need help with this question.
    5·2 answers
  • What is the common difference in the following arithmetic sequence? 2. 8, 4. 4, 6, 2007. 6,. –4. 8 –1. 6 1. 6 4. 8.
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!