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Find the equation of the parabola given the endpoints of the latus rectum at (-4,1) and (2,1)?

Answer 1

#"see explanation"#

#" the endpoints both have the same y-coordinate"# #"indicating the latus rectum is parallel to the x-axis and"# #"perpendicular to the principal axis"#

#"thus the parabola is vertical opening up or down"# #"with equation"#

#(x-h)^2=+-4a(y-k)#

#"where "(h,k)" are the coordinates of the vertex"#

#"the focus is at the midpoint of the latus rectum"#

#=(-1,1)larrcolor(blue)"coordinates of focus"#

#"the vertex is either above/below the focus"# #"depending on which way it opens"#

#"the latus rectum "=4a#

#"where a is the distance from the vertex to the focus"#

#"hence "4a=+-6rArra=+-3/2#

#"the vertex is either "(-1,1-3/2)=(-1,-1/2)#

#"or "(-1,1+3/2)=(-1,5/2)#

#(x+1)^2=6(y+1/2)larrcolor(red)"opening up"# graph{(x+1)^2=6(y+1/2) [-10, 10, -5, 5]}

#(x+1)^2=-6(y-5/2)larrcolor(red)"opening down"# graph{(x+1)^2=-6(y-5/2) [-10, 10, -5, 5]}

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.