S and S′ are the foci of the ellipse 18x2+9y2=1, and P be a point on the ellipse, then min(SP′S′)+max(SP′S′) is equal to:
Explanation
18x2+9y2=1
a2x2+b2y2=1
On comparison, we get
a^2=18,b^2=9\Rightarrow a>b

PS+PS′=2×32
b2=a2(1−e2)=18(1−e2)
⇒e=21
% Directrix
Directrix: x=2132=6
% SP and S'P Products
SP=a(1−ecosθ)
S′P=a(1+ecosθ)
SP⋅S′P=a2(1−e2cos2θ)=18(1−2cos2θ)
% Max/Min Values
(SP⋅S′P)max=18 if cosθ=0
(SP⋅S′P)min=9 if cosθ=1
Sum=18+9=27
Explanation
18x2+9y2=1
a2x2+b2y2=1
On comparison, we get
a^2=18,b^2=9\Rightarrow a>b

PS+PS′=2×32
b2=a2(1−e2)=18(1−e2)
⇒e=21
% Directrix
Directrix: x=2132=6
% SP and S'P Products
SP=a(1−ecosθ)
S′P=a(1+ecosθ)
SP⋅S′P=a2(1−e2cos2θ)=18(1−2cos2θ)
% Max/Min Values
(SP⋅S′P)max=18 if cosθ=0
(SP⋅S′P)min=9 if cosθ=1
Sum=18+9=27