Question #6c148

1 Answer
Jan 11, 2017

Answer:

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#