Find the equation of lines to an ellipse

[5ymmetrica1]

Homework Statement

Find the equation of the tangent, and the line which passes through the origin to the point where x= 3/2

4x²+9y² = 36

Homework Equations

The Attempt at a Solution

Using implicit differentiation

d/dy [4x²+9y²] =36

8x+18y dy/dx = 0

dy/dx = -8x/18y

Coordinates are (1.5, 1.732)

dy/dx = -8(1.5)/18(1.732) = -0.385

y-1.732/x-1.5 = 1/0.385

y= x- 1.5 +1.732 (equation of tangent)

But I'm stuck now how to get the line that passes through the origin to the point where x=1.5

Thanks in advance for any replies!

 

[SteamKing]

If x = 3/2, what point or points does this value of x determine on the ellipse? Can't you figure out the equation of a line if you know 2 points on that line?

 

[Simon Bridge]

How do you find the equation of a line between two points?

note: how many points on the ellipse have x=3/2?

 

[5ymmetrica1]

So if I want it to pass through the origin I can find the slope by using

1.732/1.5 = 1.155

then I can use the equation y = ax+b
where
a = 1.155
and b = 0 (y-intercept)

to give me y = 1.155x+0

does this look correct?

 

[5ymmetrica1]

So that problem worked out fine,
But the next question asks me to find the equation to a tangent of ax²+by²= c at the point where x = d

So how would I approach this?

Ive got as far as...

d/dy [ax²+by²] = c

dy/dx = -2x/2y = -x/y

by now I need the coordinates, I know that x-coordinate = d, but what would the y-coordinate be?

 

[SteamKing]

Hello. Substitution.

 

[5ymmetrica1]

what am I substituting though?

Do I substitute with f(x) or dy/dx?

 

[SteamKing]

x = d.