High school mathematics (polar coordinates)

The polar coordinates of (1) point (x, y) are expressed as:

x=rcosθ，y=rsinθ

Into the elliptic equation x 2/a 2+y 2/b 2 =1:

(rcosθ)^2 /a^2+(rsinθ)^2/b^2= 1

Suppose the polar coordinates of point A are (r 1cosθ 1, r 1sinθ 1), and the polar coordinates of point B are (r2cosθ2, r2sinθ2).

Elliptic equation r12 [(cos θ1) 2/A2+(sin θ1) 2/B2] =1,R2 2 [(cos θ 2) 2/a 2+ (.

1/OA^2+ 1/OB^2

= 1/r 1^2+ 1/r2^2

=[(cosθ 1)^2/a^2+(sinθ 1)^2/b^2]+[(cosθ2)^2/a^2+(sinθ2)^2/b^2]

=[(cosθ 1)^2+(cosθ2)^2]/a^2+[(sinθ 1)^2+(sinθ2)^2]/b^2

(cosθ2)^2=(sinθ 1)^2,(sinθ2)^2=(cosθ 1)^2

1/oa^2+ 1/ob^2= 1/a^2+ 1/b^2

(2) △AOB area = 1/2r 1r2

(r 1^2)(r2^2)

= 1/{[(cosθ 1)^2/a^2+(sinθ 1)^2/b^2][(cosθ2)^2/a^2+(sinθ2)^2/b^2]}

=(a^4)(b^4)/[b^2(cosθ 1)^2+a^2(sinθ 1)^2][b^2(cosθ2)^2+a^2(sinθ2)^2]

=(a^4)(b^4)/[b^4(cosθ 1)^2(sinθ 1)^2+a^2b^2(cosθ 1)^4+a^2b^2(sinθ 1)^4+a^4(cosθ 1)^2(sinθ 1)^2]

=(a^4)(b^4)/[b^4(cosθ 1)^2(sinθ 1)^2+a^2b^2-2a^2b^2(cosθ 1)^2(sinθ 1)^2+a^4(cosθ 1)^2(sinθ 1)^2]

=(a^4)(b^4)/[a^2b^2+(a^2-b^2)^2(cosθ 1)^2(sinθ 1)^2]

=(a^4)(b^4)/[a^2b^2 +( 1/4)(a^2-b^2)^2(sin2θ 1)^2]

When |sin2θ 1| is the largest, the area of △AOB is the smallest, θ 1=45 degrees.

(r 1^2)(r2^2)=(a^4)(b^4)/[a^2b^2 +( 1/4)(a^2-b^2)^2)，

△AOB area = a 2b 2/(a 2+b 2)

When |sin2θ 1| is the smallest, the area of △AOB is the largest, θ 1=0 degrees.

(r 1^2)(r2^2)=(a^4)(b^4)/[a^2b^2]=a^2b^2，

△AOB area =( 1/2)ab