Find the equation of lines to an ellipse

• 5ymmetrica1
In summary, the conversation discusses finding the equation of the tangent and a line passing through a given point on an ellipse. It involves using implicit differentiation and the equation of a line to solve for the desired equations.
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

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!

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?

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

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

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?

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?

Hello. Substitution.

what am I substituting though?

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

x = d.

1. How do you find the equation of a line to an ellipse?

To find the equation of a line to an ellipse, you must first determine the coordinates of the point where the line touches the ellipse. Then, you can use the slope formula to find the slope of the line. Finally, you can plug in the coordinates and slope into the point-slope form of a line to get the equation.

2. What is the difference between the equation of a line to a circle and a line to an ellipse?

The main difference between the equation of a line to a circle and a line to an ellipse is that a circle has a constant radius, while an ellipse has two different radii: the major and minor axis. This means that the slope of a line to an ellipse will change as it moves along the curve, while the slope of a line to a circle will remain constant.

3. Can you find the equation of a line to any point on an ellipse?

Yes, you can find the equation of a line to any point on an ellipse. However, the equation will vary depending on the specific coordinates of the point and the size and orientation of the ellipse.

4. How does the eccentricity of an ellipse affect the equation of the line to the ellipse?

The eccentricity of an ellipse, which is a measure of how elongated or circular the ellipse is, will affect the equation of the line to the ellipse by changing the slope of the line. The more elongated the ellipse is, the steeper the slope will be.

5. Can the equation of a line to an ellipse be written in standard form?

Yes, the equation of a line to an ellipse can be written in standard form, which is in the form of Ax + By = C. However, the coefficients for A and B will vary depending on the specific coordinates of the point and the size and orientation of the ellipse.

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