High school math: in the plane right-angle coordinate system xoy, moving point P to the two points (-√3,0), (√3,0) distance sum is equal to 4, set the trajectory of the point P for the curve C.

Solution:

(1)

According to the question C is an ellipse

2a=4

a=2

c=√3

b=1

C:x?/4+y?=1

(2)

l can't be coincident with the x-axis, otherwise AOB doesn't form a triangle

Set l:ky=(x+1)

Then x=ky-1

Associate with the elliptic equation

(k?+4)y?-2ky-3=0

From Vedder's theorem

y1+y2=2k/(k?+4)

y1y2=-3/(k?+4)

( y1-y2)?

=(y1+y2)? -4y1y1

=16(k?+3)/(k?+4)?

=16(k?+3)/[(k?+3)? +2(k?+3)+1]

=16/[(k?+3)+2+1/(k?+3)]

k?+3+1/(k?+3) is a logarithmic function that takes a minimum at k?+3=1

But since k?+3 ≥ 3

so (y1-y2)?

=16/[(k?+3)+2+1/(k?+3)]

≤16/[2+3+1/3]

|y1-y2|≤√3

So the area is maximal when l is perpendicular to the x-axis, which is 1*√3/2 = √3/2

If you still have any doubts, feel free to pursue. Wish: good luck in your studies!