Plotting the diagram of given data we get,
We know that equation of chord joining& A ( θ 1 ) and B ( θ 2 ) i s g i v e n b y , < / p > < p > \ddfrac x a cos ( θ 1 − θ 2 2 ) − y b sin ( θ 1 + θ 2 2 ) = = cos ( θ 1 + θ 2 2 ) < / p > < p > N o w g i v e n t h i s c h o r d i s p a s s i n g t h r o u g h f o c u s ( a e , 0 ) A \left({θ}_{1}\right) \text{ and } B \left({θ}_{2}\right)~is given by,</p><p>\ddfrac{x}{a}\cos \left(\dfrac{{θ}_{1} - {θ}_{2}}{2}\right) - \dfrac{y}{b}\sin \left(\dfrac{{θ}_{1} + {θ}_{2}}{2}\right) ==\cos \left(\dfrac{{θ}_{1} + {θ}_{2}}{2}\right)</p><p>Now given this chord is passing through focus~ \left(ae, 0\right) A ( θ 1 ) and B ( θ 2 ) i s g i v e nb y , < / p >< p > \ddfrac x a cos ( 2 θ 1 − θ 2 ) − b y sin ( 2 θ 1 + θ 2 ) == cos ( 2 θ 1 + θ 2 ) < / p >< p > N o w g i v e n t hi sc h or d i s p a ss in g t h ro ug h f oc u s ( a e , 0 )
\dfrac{ae}{a}cos ( θ 1 − θ 2 2 ) = cos ( θ 1 + θ 2 2 ) \cos \left(\dfrac{{θ}_{1} - {θ}_{2}}{2}\right) =\cos \left(\dfrac{{θ}_{1} + {θ}_{2}}{2}\right) cos ( 2 θ 1 − θ 2 ) = cos ( 2 θ 1 + θ 2 )
⇒ e c o s ( θ 1 − θ 2 2 ) = cos ( θ 1 + θ 2 2 ) < / p > < p > ⇒ ecos \left(\dfrac{{θ}_{1} - {θ}_{2}}{2}\right) =\cos \left(\dfrac{{θ}_{1} + {θ}_{2}}{2}\right)</p><p> ⇒ ecos ( 2 θ 1 − θ 2 ) = cos ( 2 θ 1 + θ 2 ) < / p >< p > a a 2 + b 2 \dfrac{a}{a^{2} + b^{2}} a 2 + b 2 a < / p > < p > </p><p> < / p >< p > a c o s ( θ 1 − θ 2 2 ) = a 2 + b 2 acos\left(\dfrac{{θ}_{1} - {θ}_{2}}{2}\right) = \sqrt{a^{2} + b^{2}} a cos ( 2 θ 1 − θ 2 ) = a 2 + b 2 ( θ 1 + θ 2 2 ) < / p > < p > N o w c o m p a r i n g w i t h \left(\dfrac{{θ}_{1} + {θ}_{2}}{2}\right)</p><p>Now comparing with~ ( 2 θ 1 + θ 2 ) < / p >< p > N o w co m p a r in g w i t h < / p > < p > W e g e t </p><p>We get~ < / p >< p > W e g e t < / p > < / b l o c k q u o t e > </p></blockquote> < / p >< / b l oc k q u o t e >
