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
alex41 [277]
3 years ago
15

consider the circle x^2+y^2-6x-8y=0. a.) find an equation of the tangent line to the circle at the point (0,0). b.) find the cen

ter and the radius of the circle (change into (x-a)^2+(y-b)^2=r^2 where (a,b) are the coordinates of the center and r is the radius. c.) find an equation of the tangent line to the circle at the point (6,0). d.) find the coordinates of the point where the two tangent lines intersect.

Mathematics
1 answer:
inna [77]3 years ago
4 0

The slope of the tangent line of the circle  x^2+y^2-6x-8y=0 is \frac{dy}{dx}:

to find it we use implicit differentiation:

\frac{dy}{dx}(x^2)+\frac{dy}{dx}(y^2)-\frac{dy}{dx}(6x)-\frac{dy}{dx}(8y)=0\\\\2x+2y \cdot\frac{dy}{dx}-6-8\frac{dy}{dx}=0\\\\\frac{dy}{dx}(2y-8)=-2x+6\\\\\frac{dy}{dx}= \frac{-2x+6}{2y-8}= \frac{-x+3}{y-4}

thus the slope of the tangent line at a point (x, y) of the circle is:

m= \frac{dy}{dx}=\frac{-x+3}{y-4}


part a:

m at (0, 0) is (-0+3)/(0-4)=3/(-4)=-3/4

the equation of the tangent line is

(y-0)=(-3/4)(x-0)

y=(-3/4)x


part b)

The equation of the circle can be written in standard form by completing the square:

x^2+y^2-6x-8y=0\\\\x^2-6x+y^2-8y=0\\\\(x^2-6x+9)-9+(y^2-8y+16)-16=0\\\\(x-3)^2+(y-4)^2=5^2


thus the circle has radius (3, 4) and radius 5.


part c.

m=\frac{-x+3}{y-4}=\frac{-6+3}{0-4}= \frac{-3}{-4}= \frac{3}{4}

the equation of the line is:

y-0=(3/4)(x-6)

y=(3/4)x-9/2


d) the lines are y=(-3/4)x   and   y=(3/4)x-9/2

they meet at x:

(-3/4)x=(3/4)x-9/2

(-6/4)x=-9/2

(6/4)x=9/2

(2/2)x=3/1

x=3, 

at x=3, y=(-3/4)x=(-3/4)*3=-9/4


Check the graph, generated using desmos.com

You might be interested in
100 POINTS!!!!!
Oksanka [162]

Answer:

a. Ratio of their circumference = 2:3

b. The ratio of their surface areas = 2:3

c. The ratio of their volume = 4:9

Step-by-step explanation:

Let the cylinders be labelled as A and B respectively.

Diameter of cylinder A = 6 in,

its radius = \frac{diameter}{2}

                = \frac{6}{2} = 3 in

Diameter of cylinder B = 9 in,

its radius = \frac{diameter}{2}

               = \frac{9}{2}  = 4.5 in

Thus,

a. circumference of a circle = 2\pir

Circumference of cylinder A = 2\pir

                                               = 2 x \pi x 3

                                            = 6\pi in

Circumference of cylinder B = 2\pir

                                              = 2 x \pi x 4.5

                                              = 9\pi

Ratio of their circumference = \frac{circumference of cylinder A}{circumference of cylinder B}

                                               = \frac{6\pi }{9\pi }

                                               = 2:3

b. The surface area of a cylinder = 2\pirh

the surface area of cylinder A = 2\pirh

                                                  = 6\pih

the surface area of cylinder B = 2\pirh

                                                = 9\pih

The ratio of their surface areas = \frac{surface area of cylinder A}{surface area of cylinder B}

                                                 = \frac{6\pi h}{9\pi h}

                                                 = 2:3

c. Volume of a cylinder = \pir^{2}h

volume of cylinder A = \pir^{2}h

                                = \pi x 3^{2} x h

                                = 9\pih

volume of cylinder b = \pir^{2}h

                                    = \pi x 4.5^{2} x h

                                    = 20.25\pih

The ratio of their volume = \frac{volume of cylinder A}{volume of cylinder B}

                                         = \frac{9\pi h}{20.25\pi h}

                                         = 4:9

6 0
3 years ago
How many ounces are in 1 pint?
Elenna [48]
16 fluid ounces ywww !!!
4 0
3 years ago
Read 2 more answers
How to solve this equation ​
yawa3891 [41]

Answer:

There were 15 fans that were wearing hats

Step-by-step explanation:

4 0
3 years ago
A survey of 75 students was conducted to find out where the seventh grade wanted to go to visit College colleges and universitie
Nat2105 [25]
For this question, we need to find the unit rate. 7.5 is 10% of 75 (which we can find by dividing 75 by 10). Then, in order to get 30 from this, we would have to multiply 7.5 x 4. Because we multiplied this by 4, that hints to us that 40% of 75 is 30. With this knowledge, we can find 40% of 300, which when doing the math would be 120. Therefore, because 30/75 students would want to go to Clemson University, that means 120 students out of 300 would want to go to Clemson University. Thus, our answer is 120 students
4 0
2 years ago
I need help please and thank you!​
ziro4ka [17]

Answer:

<u>v</u><u> </u><u>is</u><u> </u><u>1</u><u>9</u><u>8</u><u> </u><u>m</u><u>/</u><u>s</u>

Step-by-step explanation:

v =  \sqrt{gh}  \\ v =  \sqrt{9.8 \times 4000}  \\ v = 198 \: m {s}^{ - 1}

8 0
3 years ago
Other questions:
  • In Sal's sock drawer, there are 2 red, 2 blue, 2 yellow, 2 black, and 2 green socks. If he randomly picks one sock from the draw
    15·1 answer
  • Three sisters went shopping for Mother’s Day. Each sister bought a gift for their mom. Maggie spent 3 times as much as Karen. Ka
    6·1 answer
  • A VHS tape holds at most 360 minutes. A tape already has a 120 minute movie on it. How many 30 minute sitcoms can be recorded on
    6·1 answer
  • The tip of a pendulum moves through an angle of 0.38 radians. If the radius of the pendulum is 25
    9·1 answer
  • Solve using substitution: 2x−3y=−1 y=x−1
    15·1 answer
  • Evaluate each expression for X =
    6·1 answer
  • Find the surface area of the cone in terms of π.
    11·2 answers
  • Explain the error shown in the problem
    7·1 answer
  • After the first round of a trivia game, Ryleigh’s score is -259 points. During the second round of the game, she earns 348 point
    10·1 answer
  • PLEASE HELP!!! 50 POINTS!!!!!
    15·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!