Question

HELP HELP PLEASE. find the tangents of the curve 2x62 - 3y^2 + 12x - 12y + 22 = 0 from (7, 6). the answers are 2x - 3y + 4 and 22x - 27y +8 = 0

Answer 1

what i did was i first get the dy/dx of the equation and it was -12-4x / -12-6y..

Answer 2

then equate it to the slope form. or the m= y-y1/ x-x1..

Answer 3

@phi

Answer 4

you can simplify this
dy/dx = -12-4x / -12-6y
\[ \frac{dy}{dx}= \frac{4(x+3)}{6(y+2)} = \frac{2(x+3)}{3(y+2)} \]
now do the usual, find the slope between the point (x,y) and (7,6)
\[ \frac{2(x+3)}{3(y+2)} = \frac{y-6}{x-7}\]

Answer 5

typed it wrong. it should be 2x^2.. not 2x62.. oh my @phi thank you so much!!!!!

Answer 6

yes, I noticed (because I make the typo often enough)

Answer 7

got stuck up after cross multiplying..

Answer 8

2x^2 - 8x - 42= 3y^2 - 12y -36..

Answer 9

but you do have some nasty algebra to do...
\[ 2(x+3)(x-7) = 3(y+2)(y-6) \\
2x^2 -8x -42 = 3y^2 -12y -36 \\
(2x^2 - 3y^2) -8x -42 +12y+36=0

\]

Answer 10

now we use this trick
\[ 2x^2 - 3y^2 + 12x - 12y + 22 = 0 \\2x^2 - 3y^2= -12x+12y-22
\]

Answer 11

we can get rid of the 2x^2 - 3y^2, and solve for x

Answer 12

what do you mean??

Answer 13

get rid of the 2 terms? why?

Answer 14

\[ (2x^2 - 3y^2) -8x -42 +12y+36=0
\]
replace the stuff in parens with
\[ 2x^2 - 3y^2= -12x+12y-22 \]
to get
\[ -12x+12y-22-8x -42 +12y+36=0 \]
simplify that mess and solve for x

Answer 15

if you can get rid of quadratics, then you should do so...

Answer 16

i didnt derive a quadratic..

Answer 17

its -20x + 24y - 28

Answer 18

you mean -20x + 24y - 28=0
divide by 4
-5x + 6y -7=0
x= (6y-7)/5

now use that value of x in our cross product:
\[ 2(x+3)(x-7) = 3(y+2)(y-6) \]
and we will get a quadratic in y
solve for y... you get 2 values. find x using x= (6y-7)/5
find the slope.
and then the line

Answer 19

2 (6y-7/5) equals to?? 12y - 14/5? am i rght?

Answer 20

\[ 2(x+3)(x-7) = 3(y+2)(y-6) \\
2\left( \frac{(6y-7)}{5} + 3\right)\left( \frac{(6y-7)}{5} -7\right)= 3(y+2)(y-6)
\]

Answer 21

this is so algebra okay..

Answer 22

ill answer it

Answer 23

:)

Answer 24

((12y+4/5) +6) ((6y-7)/5 -7)= 3y^2 - 12y - 12..

Answer 25

( (12y-14)/5 +6) ((6y-7)/5 -7)= 3y^2 - 12y - 36

Answer 26

(12y -14/5) (6y-7/5) = 72y^2 - 84y - 84y + 98 / 25??

Answer 27

my answer is 72y^2 - 84y - 84y + 98/ 25 - 84y - 98/5 + 36y- 42/5 - 42.. am i right? o did i missed a thing? a sign?

Answer 28

72y^2 - 84y/5 - 84y/5 + 98 / 25

or
(12y -14/5) (6y-7/5) = 72y^2 -168y/5 + 98/25

I did it differently..
\[ 2\left( \frac{(6y-7)}{5} + 3\right)\left( \frac{(6y-7)}{5} -7\right)= 3(y+2)(y-6) \\
2\left( \frac{(6y-7+15)}{5} \right)\left( \frac{6y-7-35}{5} \right)= 3(y+2)(y-6) \\
2\left( \frac{6y+8}{5} \right)\left( \frac{6y-42}{5} \right)= 3(y+2)(y-6) \\
2\cdot 2\left( \frac{3y+4}{5} \right)\cdot 6 \cdot \left( \frac{y-7}{5} \right)= 3(y+2)(y-6) \\
24 \frac{(3y+4)(y-7)}{25}= 3(y^2 -4y -12) \\
8 (3y+4)(y-7)= 75(y^2 -4y -12)
\]

Answer 29

the very last line in the above post should have 25 on the right side (not 75)
I divided both sides by 3
\[ 8 (3y+4)(y-7)=25(y^2 -4y -12) \\
8 (3y^2 -17y-28) =25y^2 - 100y -300 \\
24y^2 - 136y -224= 25y^2 - 100y -300
\]
now simplify

Answer 30

wait. can i ask you about how did you derived 8?

Answer 31

got it now. 24/3

Answer 32

yes. and then multiply both sides by 25 ... and I forgot I did 3/3 on the right side

Answer 33

i got 2 and - 38. FINALLY HOO. THANKS TO YOOOOOOU

Answer 34

still some work to do... but the nasty stuff is over with...

Answer 35

can i ask you something? about getting the dy/dx of xy=18?

Answer 36

xy=18

you can solve for y
\[ y = 18 x^{-1} \]
\[ \frac{dy}{dx} = -18x^{-2} \]

Answer 37

raised to -1. why??

Answer 38

or use implicit differentiation
x dy/dx + y dx/dx = 0
dy/dx = -y/x

if you like, you can use y = 18/x and get
dy/dx = -18/x^2

Answer 39

raised to -1. why??

\[ \frac{1}{x}= x^{-1} \]
right ?
taking the derivative of a power is easy, so I change 1/x to x^(-1)

Answer 40

squared?

Answer 41

are you asking how to take the derivative of x^n ?
\[ d x^n = n x^{n-1} dx \]

Answer 42

okay!! got it now..

Answer 43

thank you so much for the help, patience. yes and everything. need to finish my assignment. 2 more items to go. hope i can do this :) God bless you..