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
grigory [225]
3 years ago
12

Verify that the given point is on the curve and find the lines that are (a) tangent and (b) normal to the curve at the given poi

nt.
I. x2 + xy - y2 = 1, (2,3) (10)
II x2 + y2 = 25, (3,-4)
III. x3y2 = 9, (-1,3)
IV. y2 – 2x – 4y - 1 = 0, (-2, 1)
Mathematics
1 answer:
lara [203]3 years ago
3 0

For each curve, plug in the given point (x,y) and check if the equality holds. For example:

(I) (2, 3) does lie on x^2+xy-y^2=1 since 2^2 + 2*3 - 3^2 = 4 + 6 - 9 = 1.

For part (a), compute the derivative \frac{\mathrm dy}{\mathrm dx}, and evaluate it for the given point (x,y). This is the slope of the tangent line at the point. For example:

(I) The derivative is

x^2+xy-y^2=1\overset{\frac{\mathrm d}{\mathrm dx}}{\implies}2x+x\dfrac{\mathrm dy}{\mathrm dx}+y-2y\dfrac{\mathrm dy}{\mathrm dx}=0\implies\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{2x+y}{2y-x}

so the slope of the tangent at (2, 3) is

\dfrac{\mathrm dy}{\mathrm dx}(2,3)=\dfrac74

and its equation is then

y-3=\dfrac74(x-2)\implies y=\dfrac74x-\dfrac12

For part (b), recall that normal lines are perpendicular to tangent lines, so their slopes are negative reciprocals of the slopes of the tangents, -\frac1{\frac{\mathrm dy}{\mathrm dx}}. For example:

(I) The tangent has slope 7/4, so the normal has slope -4/7. Then the normal line has equation

y-3=-\dfrac47(x-2)\implies y=-\dfrac47x+\dfrac{29}7

You might be interested in
2+6i/2-4i = a + bi What is the value of a? What is the value of b?
Talja [164]

\dfrac{2+6i}{2-4i}\\\\\\=\dfrac{2(1+3i)}{2(1-2i)}\\\\\\=\dfrac{1+3i}{1-2i}\\\\\\=\dfrac{(1+3i)(1+2i)}{(1-2i)(1+2i)}\\\\\\=\dfrac{1+2i+3i+6i^2}{1-(2i)^2}\\\\\\=\dfrac{1+5i-6}{1+4}~~~~;[i^2=-1]\\\\\\=\dfrac{5i-5}{5}\\\\\\=\dfrac{5(i-1)}5\\\\=i-1\\\\=-1+i\\\\\text{It is in a form of a +bi so,}~ a = -1~ \text{and}~ b = 1

6 0
2 years ago
A map of a park is created on a coordinate plane. Each until on the map represents 1 kilometer. A tennis court is located at (0,
pogonyaev

Answer:

Jul 11, 2016 · - Gush Tela, a 54-year-old man who was forced to flee with his family after the town of Humera, located in Ethiopia's Tigray region, was shelled by the Ethiopian Army. He was describing the scene he encountered as he returned to the town several days later; he and his family are now in a refugee camp in Sudan (as quoted on The Guardian )

Step-by-step explanation:

7 0
3 years ago
What are some quotations for 36
gladu [14]
These are some quotations that I don’t know
8 0
3 years ago
Tristan is building steps up to a deck that is 156 5 6 yards above the ground. The steps must all be the same height. If he want
ss7ja [257]

Answer:

The height of each step is 1957 yard

Step-by-step explanation:

we are given

Tristan is building steps up to a deck that is 15656 yards above the ground

so, total height =15656 yards

Number of steps =8

now, we can use formula

Height of each step = ( total height)/( number of steps)

now, we can plug these values

and we get

Height of each step is

=\frac{15656}{8}yards

=1957yards

8 0
3 years ago
Lane's arm span is 61 inches. If his height is 1⁄4 foot longer than his arm span, how tall is he? feet and inches
melamori03 [73]
If lane's are span is 61 inches. his height is 1/4 foot long than his arm span. his height is <span>5 ft 3 inches </span> 
6 0
3 years ago
Other questions:
  • A mixture can be classified as a solution, suspension, or colloid based on _____. its total number of particles
    13·2 answers
  • Let f(x) = x2 and g(x) = x − 3.
    12·1 answer
  • Kelly went to the store 13 times last month. She buy 7 peppermints and 5 swedish fish every time she goes to the store. How much
    13·2 answers
  • Give an example of an irrational number that is greater than 0
    9·1 answer
  • What is the absolute value off -37
    12·2 answers
  • I did this and turned it in, I got a C on it, could you tell me what to fix so I can get an A?
    13·1 answer
  • 2 to the power of 4 - 3+2x5
    14·1 answer
  • The spinner is divided into 5 equal sections. Trina is going to spin the arrow once.
    12·1 answer
  • What is differentiation....​
    10·2 answers
  • A ring is reduced to £840. This is a saving of 40% of the original price. What was the original price
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!