Question

The graph of the equation X^2 - XY + Y^2 = 9 is the rotated ellipse. Find the equation of the lines tangent to the ellipse at the two points at which it intersects the x axis, and show that these lines are parallel. The graph of the equation X^2 - XY + Y^2 = 9 is the rotated ellipse. Find the equation of the lines tangent to the ellipse at the two points at which it intersects the x axis, and show that these lines are parallel. @Mathematics

Answer 1

my dy/dx = -2x+y/(2y-x)

the two equation i got is y=2x-6 and y=2x+6

Answer 2

how do i show that these line are parrallel

Answer 3

Two lines that have the same slope and different intersects are parallel by definition. But if you'd like, you can set the equations equal to each other and show that there is no solution, which means they never intersect, which is another working definition of parallel lines.

Answer 4

if their slopes are the same, they must be parallel and must not intercept.

Answer 5

*different intercepts*

Answer 6

if i should work then its 0=12 means no solution

Answer 7

intersect*

Answer 8

@moneybird we should probably learn to type at some point, huh? :)

Also, yeah. That means there is no solution, i.e. no point where the two lines cross.

Answer 9

you guys think the 2 equation i got is right?

Answer 10

I'm a bit confused. You said "my dy/dx = -2x+y/(2y-x)." That's a nonlinear differential equation...

Answer 11

:)

Answer 12

i use implicit differentiation to find that

Answer 13

\[x^2-xy+y^2=9\]\[\frac{d}{dx}\left [ x^2-xy+y^2=9 \right ]\]\[2x-y\frac{dy}{dx}+2y\frac{dy}{dx}=0\]\[\frac{dy}{dx}+2x=0\]\[\frac{dy}{dx}=-2x\]

Answer 14

I left out the y; it should be\[\frac{dy}{dx}=-\frac{2x}{y}\]

Answer 15

product rules for -XY?

Answer 16

2X -Y-Xdy/dx + 2Ydy/dx=0