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]
3 years ago
7

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

Mathematics
1 answer:
algol [13]3 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
Which statement about the dilation of these triangles is true?
natulia [17]

Jabardasth question is the scale factor is 2

7 0
3 years ago
Read 2 more answers
HELP I’M GONNA FAIL!!!!!! Pls help ASAP! I NEED TO PROMOTEEEEEE!!!!
Nana76 [90]
Volume of sphere is
V=4/3pi r^3

Where r is 2 in this example so,

V= 10.67 pi
V= 33.51
7 0
2 years ago
Read 2 more answers
How many different linear arrangements are there of the letters a, b,c, d, e for which: (a a is last in line? (b a is before d?
inna [77]
A) Since a is last in line, we can disregard a, and concentrate on the remaining letters.
Let's start by drawing out a representation:

_ _ _ _ a

Since the other letters don't matter, then the number of ways simply becomes 4! = 24 ways

b) Since a is before d, we need to account for all of the possible cases.

Case 1: a d _ _ _ 
Case 2: a _ d _ _
Case 3: a _ _ d _
Case 4: a _ _ _ d

Let's start with case 1.
Since there are four different arrangements they can make, we also need to account for the remaining 4 letters.
\text{Case 1: } 4 \cdot 4!

Now, for case 2:
Let's group the three terms together. They can appear in: 3 spaces.
\text{Case 2: } 3 \cdot 4!

Case 3:
Exactly, the same process. Account for how many times this can happen, and multiply by 4!, since there are no specifics for the remaining letters.
\text{Case 3: } 2 \cdot 4!

\text{Case 4: } 1 \cdot 4!

\text{Total arrangements}: 4 \cdot 4! + 3 \cdot 4! + 2 \cdot 4! + 1 \cdot 4! = 240

c) Let's start by dealing with the restrictions.
By visually representing it, then we can see some obvious patterns.

a b c _ _

We know that this isn't the only arrangement that they can make.
From the previous question, we know that they can also sit in these positions:

_ a b c _
_ _ a b c

So, we have three possible arrangements. Now, we can say:
a c b _ _ or c a b _ _
and they are together.

In fact, they can swap in 3! ways. Thus, we need to account for these extra 3! and 2! (since the d and e can swap as well).

\text{Total arrangements: } 3 \cdot 3! \cdot 2! = 36
7 0
3 years ago
Please help me out please
mariarad [96]

Answer:

1.) \\625^2 Ft.\\\\2.)\\306^2Ft.\\\\3.)\\18,750^2 Ft.\\

Step-by-step explanation:

1.) \\25*25=625\\\\2.) \\(103*2)+(50*2)=\\206+100=306\\\\3.) \\750*25=18,750

8 0
3 years ago
A transformation of rectangle LMNO results in rectangle L'M'N'O'.
Helen [10]

Answer:

As you can see Rectangle L M NO and Rectangle L'M'N'O' are such that

1. Size of Rectangle L M NO > Size of Rectangle L'M'N'O'

2. Sides of, means length and breadth of Rectangle L M N O is not in proportion to Rectangle L'M'N'O'. i.e

    \frac{LM}{L'M'}=\frac{8}{6}=\frac{3}{2} \\\\\frac{LO}{L'O'}= \frac{4}{4}=1

3. Out of four possibilities given :

, Reflection and Rotation is not possible because in all the three cases Pre-image and image are not congruent or similar.

As in,Rectangle LMNO →when length and breadth get reduced by scale of 2 and 0 and Dilation by scale factor of< 1  we get Rectangle L'M'N'O'.Centre of dilation may be other than origin.

So, Stretching of preimage i.e Rectangle L M NO→ means Shrinking has taken place to get image Rectangle L'M'N'O'.As shrinking is not an option.

So, Dilation by a factor less than 1, center of dilation other than origin has taken place.



8 0
3 years ago
Read 2 more answers
Other questions:
  • Sam ran at 6km/h for 2h 20 mins. What distance did he run?
    14·2 answers
  • Can someone pls help me with the question? its in the picture :)
    14·2 answers
  • Which statement is true?
    13·2 answers
  • Kelsey buys 4 packages of juice boxes for a class party. Each pack has 1 point 5 rows with 2 juice boxes in each row. How many b
    9·1 answer
  • Round 8,088 to the nearest thousand
    5·2 answers
  • (3x+1)/(3x+2) = (x+2)/(x-5)
    7·1 answer
  • — 3х + Зу = - 24<br> how do i convert to slope intercept form
    8·1 answer
  • Can someone solve this for me pls
    6·1 answer
  • In the middle school band, there are 5 flutes for every 2 tubas. Write the ratio of flutes to tubas in three different ways.
    13·2 answers
  • Can someone help please my sister needs it though with the whole thing
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!