Question

Question #6c148

Answer 1

The midpoint of the chord will start at one of the two points where the slope of the tangent line is equal to 1 and end at the other point where the slope of the tangent line is equal to 1.

Explanation:

Therefore, we need to find the two points on the ellipse where the slope of the tangent line is equal to 1.

Implicitly differentiate the given equation:

`(d(x^2))/dx + (d(4y^2))/dx = (d(1))/dx`

`2x + 8ydy/dx = 0`

`8ydy/dx = -2x`

`dy/dx = -x/(4y)`

Set `dy/dx` = 1:

`-x/(4y) = 1`

`y = -x/4`

Substitute `-x/4` for y into the equation for the ellipse:

`x^2 + 4x^2/16 = 1`

`5/4x^2 = 1`

`x^2 = 4/5`

`x = +-2sqrt(5)/5`

Substitute `4/5` for `x^2` into the equation for the ellipse:

`4/5 + 4y^2 = 1`

`4y^2 = 1/5`

`y^2 = 1/20`

`y = +-sqrt5/10`

The locus of the midpoint is a line segment between the points:

`(-2sqrt(5)/5, sqrt5/10) and (2sqrt(5)/5, -sqrt5/10)`

m = `(-sqrt5/10 - sqrt5/10)/(2sqrt(5)/5 - -2sqrt(5)/5) = -1/4`

The equation of the locus is the line segment:

`y = -1/4(x - 2sqrt(5)/5) - sqrt5/10; -2sqrt(5)/5< x < 2sqrt(5)/5`