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
Lisa [10]
2 years ago
7

Which points on the curve of x^2 - xy - y^2 = 5 have vertical tangent lines?

Mathematics
1 answer:
algol [13]2 years ago
8 0

We need to differentiate this with respect to x to see if we can find an expression for the derivative of y at various points.  That will be the slope of the tangent to the curve.  Then we want to see where that derivative might be infinite -- i.e., where the tangent is vertical.

 

It's not written as a function, but it can still be differentiated using the chain rule:

 

x2 + xy + y2 = 3

(2x) + (x dy/dx + y dx/dx) + (2y dy/dx) = 0

 

(I used parentheses to show the differentiation of each term in the original equation.)

 

2x + x dy/dx + y + 2y dy/dx = 0

2x + y = -x dy/dx - 2y dy/dx

2x + y = dy/dx (-x -2y)

-(2x + y)/(x + 2y) = dy/dx

 

We have the derivative of y, but it's defined partly in terms of y itself.  That's OK.  Let's go on...

 

So where would the slope be infinite?  That would happen when x + 2y = 0, or y = -x/2

 

Let's plug that in for y in the original equation to find points where that's the case.

 

x2 + xy + y2 = 3

x2 + x(-x/2) + (-x/2)2 = 3

x2 - x2/2 + x2/4 = 3

3x2 / 4 = 3

x2 = 4

x = ±2

 

So we have two x values where the tangent might be vertical.  Let's plug them into the equation and see what the y values are.  First x = 2...

 

x2 + xy + y2 = 3

4 + 2y + y2 = 3

y2 + 2y + 1 = 0

(y + 1)2 = 0

y = -1

 

So at the point (2, -1) the tangent is vertical.

 

Now try x = -2...

 

x2 + xy + y2 = 3

4 - 2y + y2 = 3

y2 - 2y + 1 =0

(y - 1)2 = 0

y = 1

 

So at the point (-2, 1) the tangent is vertical.

You might be interested in
5. In one year, there were 125 major new automobile types available. If you look at the miles per gallon (MPG) for these vehicle
ziro4ka [17]

Answer:

The answer is "99.82% and 86.99%".

Step-by-step explanation:

In point a:

=P(X < 45)\\\\= P(Z < \frac{45-29.09}{5.46}) \\\\=P(Z < 2.9139)\\\\=0.9982\\\\=99.82\%

In point b:

=P(X >17)\\\\= P(Z > \frac{17-22.37}{4.77}) \\\\=P(Z > -1.1258)\\\\=0.8699\\\\=86.99\% \\\\

5 0
2 years ago
One of the students in the class scored 100 on the midterm but got overconfident, slacked off, and scored only 15 on the final e
miss Akunina [59]

Answer:

Increase

Step-by-step explanation:

Since your not including the bad mark, it'd increase.

3 0
3 years ago
What formula do we use to find slope?
Kamila [148]

Answer:Remember, the formula for slope is (y2 - y1) / (x2 - x1).

Step-by-step explanation:

4 0
3 years ago
Read 2 more answers
-4x + 6 + 2x + 5, Solve for X
pychu [463]
-4x + 2x = -2x
6 + 5 = 11
Therefore, 11 - 2x = 0
11 = 2x
x = 11/2 = 5.5
8 0
3 years ago
a golf practice range has 245 balls. the owner bought a carton of golf ball. how many golf balls does the owner have after buyin
konstantin123 [22]
Uh i dont know are you supposed to add or substract multiply or divid use nixed numbers on no mixed numbers
5 0
3 years ago
Read 2 more answers
Other questions:
  • About 400,000 people visited an art museum in December. What could be the exact number of people who visited the art museum
    10·2 answers
  • HELP FAST PLEASE!!!!!!
    8·1 answer
  • Someone please help me
    15·1 answer
  • Find the product<br> .<br> -1(6) =<br> 1
    5·1 answer
  • What is 45+(-15) +7
    15·1 answer
  • What is this anwser for 1/3+v=25 for in this problem
    9·1 answer
  • Solve for s.<br> S-1<br> 2<br> +3<br> S
    14·1 answer
  • Test scores: Scores on a statistics exam had a mean of 75 with a standard deviation of 5. Scores on a calculus exam had a mean o
    13·1 answer
  • A bin contains 25 light bulbs, 5 of which are in good condition and will function for at least 30 days, 10 of which are partiall
    6·1 answer
  • Solve xy − 2 = k for x.
    15·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!