Question

find the tangents of the curve 2x^2 + y^2 = 54 from (10, 1) please help. the answers are x-y -9 = 0 and 53x + 73y - 603= 0

Answer 1

its just getting the dy/dx

Answer 2

Getting the dy/dx will give you the gradient of the tangent to the line at a given point. What else would you need for a tangent(straight line) equation?

Also, why are there 2 answers?

Answer 3

2 slopes

Answer 4

Ah I get it. So what is the dy/dx you get? And what does it represent?

Answer 5

the slope..

Answer 6

i got -4x/2y

Answer 7

@phi

Answer 8

First you rearrange the equation to solve for y:

\[y=\pm \sqrt{54-2x ^{2}}\]I would assume you will have 2 answers because you will get 2 slopes here due to the plus and minus. What are those slopes?

Answer 9

trying to figure it out but i can't..

Answer 10

how about the dy/dx

Answer 11

Well, now that you have the equation in terms of y you can find dy/dx. So differentiate that equation. I think you know how to do this part :)

Answer 12

Hint: you can rewrite the equation as\[y=\pm \left( 54-2x ^{2} \right)^{1/2}\]

Answer 13

@silverxx Should I show you more?

Answer 14

yes yes. i know..

Answer 15

wait is -4x/ 2y is -2x/y??

Answer 16

Yes

Answer 17

then help me to get y on this equation please hehehe -2x/y = y-1/ x- 10

Answer 18

Hmm. Not sure where I am going myself with this, but if what you meant is \[\frac{ -2x }{ y }=\frac{ y-1 }{ x-10 }\] y is \[\frac{ 1 }{2 }\left( 1-\sqrt{-8x ^{2}+80x+1} \right)\]

Answer 19

yes yes thats it

Answer 20

my solution was it turned out to be a quadratic

Answer 21

dear wolfe8
what program are you using to write your equations so well :)

Answer 22

First, I would use implicit differentiation to find dy/dx in
2x^2 + y^2 = 54

4x + 2y dy/dx =0
dy/dx= -2x/y

Answer 23

once you have the equation
\[ \frac{ -2x }{ y }=\frac{ y-1 }{ x-10 } \]
cross multiply to get
\[ -2x^2 +20x = y^2 -y\\
y+20x= 2x^2+y^2
\]
from the original equation 2x^2 + y^2 = 54
we can simplify
\[y+20x= 2x^2+y^2\\y+20x=54 \\ y= 54-20x\]

substitute for y in the original equation
\[ 2x^2+y^2=54\\
2x^2 +(54-x)^2=54
\]
that gives a hairy quadratic with roots x=3 and x= 159/67

Answer 24

to find the tangent lines, we can solve for the slope, and use the point slope form of the equation of a line.

first find y, using y = 54-20x.
with x =159/67 we get y = 438/67

the slope m= -2x/y = -318/438= - 53/73

knowing (10,1) is on the line, we have
y - 1 = (-53/73) (x-10)
-73y +73 = -53x +530
53x -73y -603=0

you can follow the same procedure using the other root x=3 to find the equation of the other tangent line.

Answer 25

phi, how do you type out the equations so clearly, they appear just like in the text books :)

Answer 26

use the equation editor (see button on lower left of input area)

Answer 27

@carlyn101 Sorry I didn't see your question. As phi said, it is the editor.

Answer 28

it is latex so you can type it directly in, using \ [ \ ] (with no spaces between the slash and bracket.
or \ ( \ ) for embedded forms. example \( x^2 \)

also, right click on an equation and select Show Math as Tex commands
if you want to copy and paste some complicated expression

Answer 29

I just typed in this --> 2x^2 + y^2 = 54 then i put it in the equation editor, then hit the INSERT button and nothing happened ?

Answer 30

Also do you know if i can do equations on a wacom writing tablet ?

Answer 31

@carlyn101

yes, open the equation editor,
type into its input area
you should see the latex form in the large box above the input area
use the buttons on the right side to select formats.. e.g. square root, integral sign, etc
when finished with the equation, select the INSERT button in the equation editor.

The equation will show up in the "input area" where you type in text. Select POST to publish the result.

Answer 32

thank you thank you @phi

Answer 33

I noticed I lost a sign at the very end...

knowing (10,1) is on the line, we have
y - 1 = (-53/73) (x-10)
-73y +73 = 53x - 530 (multiply both sides by -73, distribute +53)
53x +73y -603=0

now it matches your answer up top.

Answer 34

i have my answer already.. i didnt give up :) happy having same solution with you ;)

Answer 35

\[2x^2 + y^2 = 54\] thanks phi and wolf