Question

find the tangents of the following curves y^2 = 12x from (2, 7). the answers are 3x - y +1 = 0 and x- 2y + 12 = 0. helppp

Answer 1

calculus?

first, use implicit differentiation on y^2 = 12
can you do that?

Answer 2

2y dy/dx = 12

Answer 3

dy/dx = 12/ 2y

Answer 4

ok,
or dy/dx= 6/y

Answer 5

so dy/dx = 6/7

Answer 6

let a point on the curve be (x,y)
the slope from that point to (2,7) is
\[ m = \frac{y-7}{x-2} \]
we want this slope to match dy/dx ( the derivative is the slope)
\[ \frac{6}{y} = \frac{y-7}{x-2} \]

we could solve that if we got rid of the x.
the curve is
y^2 = 12 x and so
\[ x= \frac{y^2}{12} \]

use that value for x in our equation
\[ \frac{6}{y} = \frac{y-7}{\frac{y^2}{12}-2} \]

Answer 7

can you solve for y ?

Answer 8

sure, just wait sir :)

Answer 9

its decimal.. not whole number or i was just too stupid solving algebra..

Answer 10

what do you get after cross multiplying ?

Answer 11

6 (y^2 / 12 -2) = y (y-7)

6y^2 / 12 -12 = y^2 - 7y

6y^2 - y^2 = -7y + 12

1/2 y^2 - 7y -12 = 0..

Answer 12

am i right?? im sorry

Answer 13

6y^2 / 12 -12 = y^2 - 7y

at this step you have
\[ \frac{6}{12} y^2 - 12 = y^2 -7y \\
\frac{1}{2} y^2 - 12 = y^2 -7y
\]
I would multiply both sides by 2 to get rid of the fraction
\[ y^2 -24= 2y^2 -14y \]
can you finish ?

Answer 14

its still the same.. 15. 54 and -1.5

Answer 15

noooo

Answer 16

12 and 2

Answer 17

\[ y^2 -24= 2y^2 -14y \\
y^2 -14y + 24 = 0
\]
that factors

Answer 18

its 12 and 2

Answer 19

now that we are over that snafu...

find the slope of the tangent line using m = 6/y (from the derivative way up top)
we know (2,7) is a point on the line, so we can use the point slope form:

y - y0= m ( x - x0) where m is the slope, and (x0,y0) is a point on the line

rearrange into standard form to match your answers up top

Answer 20

slope is 6/7

Answer 21

y-7 = 6/7 (x-2)

Answer 22

no, use the y values we found from
\[ y^2 -14y + 24 = 0 \]

Answer 23

so 6/12 or 1/2

Answer 24

and 6/2 or 3

Answer 25

yes. you will now get two different equations for the 2 lines

Answer 26

got cha!! THANKS :)))))))))))