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
Flauer [41]
3 years ago
7

Find an equation of the tangent line to the curve 2(x^2+y^2)2=25(x^2−y^2) (a lemniscate) at the point (3,1)

Mathematics
1 answer:
Sholpan [36]3 years ago
6 0
\bf 2[x^2+y^2]^2=25(x^2-y^2)\qquad \qquad 
\begin{array}{lllll}
&x_1&y_1\\
%   (a,b)
&({{ 3}}\quad ,&{{ 1}})\quad 
\end{array}\\\\
-----------------------------\\\\
2\left[ x^4+2x^2y^2+y^4 \right]=25(x^2-y^2)\qquad thus
\\\\\\
2\left[ 4x^3+2\left[ 2xy^2+x^22y\frac{dy}{dx} \right]+4y^3\frac{dy}{dx} \right]=25\left[2x-2y\frac{dy}{dx}  \right]
\\\\\\
2\left[ 4x^3+2\left[ 2xy^2+x^22y\frac{dy}{dx} \right]+4y^3\frac{dy}{dx} \right]=50\left[x-y\frac{dy}{dx}  \right]
\\\\\\


\bf \left[ 4x^3+2\left[ 2xy^2+x^22y\frac{dy}{dx} \right]+4y^3\frac{dy}{dx} \right]=25\left[x-y\frac{dy}{dx}  \right]
\\\\\\
4x^3+4xy^2+4x^2y\frac{dy}{dx}+4y^3\frac{dy}{dx}+25y\frac{dy}{dx}=25x
\\\\\\
\cfrac{dy}{dx}[4x^2y+4y^3+25y]=25x-4x^3+4xy^2
\\\\\\
\cfrac{dy}{dx}=\cfrac{25x-4x^3+4xy^2}{4x^2y+4y^3+25y}\impliedby m=slope

notice... a derivative is just the function for the slope

now, you're given the point 3,1, namely x = 3 and y = 1

to find the "m" or slope, use that derivative, namely f'(3,1)=\cfrac{25x-4x^3+4xy^2}{4x^2y+4y^3+25y}

that'd give you a value for the slope

to get the tangent line at that point, simply plug in the provided values
in the point-slope form

\bf y-{{ y_1}}={{ m}}(x-{{ x_1}})\qquad
\begin{cases}
x_1=3\\
y_1=1\\
m=slope
\end{cases}\\ \qquad \uparrow\\
\textit{point-slope form}

and then you solve it for "y", I gather you don't have to, but that'd be the equation of the tangent line at 3,1

You might be interested in
Which shapes have at least 1 right angle? Choose all that are correct.​
satela [25.4K]

Answer:The Pink square, Green Rectangle, and the Orange figure :)

Step-by-step explanation:

4 0
3 years ago
Read 2 more answers
What is the probability of drawing a five then a red card?
Airida [17]
Ummm sorry cant answer that

8 0
3 years ago
4d2e - d2e find difference
Nataliya [291]
I think the answer is 1d.
7 0
3 years ago
Read 2 more answers
The output of a function in 7 more than 3 times the input. find the input when the output is -8
kiruha [24]
7+ 3x= -8 
-8-7=-15
-15 divided by 3= -5
-5 is the answer
4 0
3 years ago
I have the answer here
NISA [10]
So what do u need help on??
6 0
3 years ago
Read 2 more answers
Other questions:
  • Perimeter of a triangle with side lengths 2x, (3x-1) and (4x+5)?
    13·1 answer
  • Hi plz help me with this geometry question
    8·1 answer
  • 96 · (70 · n) = (96 · 70) · 28
    5·2 answers
  • Someone help me fast please geometry.
    10·1 answer
  • 18h^3 + 45h^2 - 8h - 20
    12·1 answer
  • Please help me if you can help me
    8·1 answer
  • Two consecutive even integers have the following property. When the smaller integer is added to three
    14·1 answer
  • Work out the equation of the line which passes through the point (1,-2)and is parallel to the point y=x+4
    9·1 answer
  • the office supplies account started the year with a 4000 debit balance. During the year, the company purchased supplies for 13,4
    7·1 answer
  • Compare these two numbers which one has a greater value <br> 8.5 x 10^23 or<br> 9.99 x 10^21
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!