find the coordinates of the midpoint of the chord in line x+2y-1=0 which intersects the ellipse: x^2+2y^2=3

Question

Can I request for the step by step solutions? Thank you so much!

Answer 1

Find the intersection of the line with the ellipse.

From the equation of the line: x = 1 - 2y

Substitute in the equation for the  ellipse: (1 - 2y)² + 2y² = 3

Expand and collect terms: 6y² - 4y -2 = 0

Factor: (y-1)(6y + 2) =0 => y = 1 and y = -1/3

When y = 1, x = -1 and when y = -1/3, x = 5/3

 

Now you have the endpoints of the chord. Draw a picture of this chord.

The distance in the x direction is 1 + 5/3 = 8/3 and the x midpoint is -1 + 4/3 = 1/3.

The distance in the y direction is 1 + 1 = 2 and the y midpoint is -1 + 1 = 0.

Answer 2

Hello Eldrian,

Here are the steps I am going to use:

1) Solve the linear equation for x (instead of y to avoid fractions :-)  ).

2) Plug this expression for x into the ellipse equation

3) Solve the new quadratic equation for the y values of points of intersection

4) Find the corresponding x values for the points of intersection

5) Find the midpoint of those points of intersection

 

 

Here we go:

1) x + 2y - 1 = 0

→ x = 1 - 2y

 

2) (1 - 2y)² + 2y² = 3

 

3) → 1 - 4y + 4y² + 2y² = 3

→ 6y² - 4y -2 = 0

→ 3y² - 2y -1 = 0

→ (3y + 1)*(y - 1) = 0

→ y = 1 and y = -1/3

 

4)

For y = 1

→ x = 1 - 2(1) = -1
→ (-1, 1) for one point of intersection

 

For y = -1/3

→ x = 1 - 2(-1/3) = 1 + 2/3 = 5/3

→ (5/3, -1/3) for the other point of intersection

 

5) For midpoint, just average the x-values separately from averaging the y-values

→ x_mpt = (5/3 + (-1)) / 2 = (2/3) / 2 = 1/3

and y_mpt = (1 + (-1/3)) / 2 = (2/3) / 2 = 1/3

 

Answer: (1/3 1/3).

 

I hope the above helps, and thank you for posting the question.

Michael Ehlers