if the line px+qy = r intersects the ellipse x2 + 4y2 = 4 in points, whose eccentric angles differ - Maths - Conic Sections

Question

if the line px+qy = r intersects the ellipse x2 +
4y2 = 4 in points, whose eccentric angles differ by
pie/3 , then prove that r2 = 3/4 (4p2​
+q2)? ? please explain what eccentric angle refer to
??

Rahul Raj · Accepted Answer

dear student

let the two points be (acosθ,bsinθ) and (acos(θ+π/3),bsin(θ+π/3))for ellipse a=2,b=1the equation of the line joining these points is(y-bsinθ)=(x-acosθ)bsin(θ+π/3)-bsinθacos(θ+π/3)-acosθputting a=2,b=1(y-sinθ)=(x-2cosθ)bsin(θ+π/3)-bsinθacos(θ+π/3)-acosθ(y-sinθ)=(x-2cosθ)3cosθ-sinθ-2(3sinθ+cosθ)simplifyingx(3cosθ-sinθ)+2y(3sinθ+cosθ)=23comparing topx+qy=rp=(3cosθ-sinθ)r23q=(3sinθ+cosθ)r3RHS=34(4p2+q2)=34(4(3cosθ-sinθ)r232+(3sinθ+cosθ)r32)=34(4(-23cosθsinθ+3cos2θ+sin2θ+23cosθsinθ+3sin2θ+cos2θ)r212)=r2=LHS

regards

if the line px+qy = r intersects the ellipse x2 + 4y2 = 4 in points, whose eccentric angles differ by pie/3 , then prove that r2 = 3/4 (4p2​ +q2)? ? please explain what eccentric angle refer to ??

if the line px+qy = r intersects the ellipse x²+ 4y² = 4 in points, whose eccentric angles differ by
pie/3 , then prove that r² = 3/4 (4p² +q²)? ? please explain what eccentric angle refer to ??