If P(θ1) and Q(θ2) are the extremities of any focal chord of the hyperbola x22023-y22023=1, then the value of cos2θ1+θ22cos2θ1−θ22=____cosθ1−θ22≠0

Equation of any chord joining the points P(θ1) and P(θ2) to x22023−y22023=1is x2023cos(θ1-θ22)-y2023sin(θ1+θ22)=cos(θ1+θ22)If it passes through focus (±4046,0), then we get ±2cos(θ1-θ22)=cos(θ1+θ22)

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If $P (θ_{1}) a n d Q (θ_{2})$ are the extremities of any focal chord of the hyperbola $\frac{x^{2}}{2023} - \frac{y^{2}}{2023} = 1$ , then the value of $\frac{\cos^{2} (\frac{θ_{1} + θ_{2}}{2})}{\cos^{2} (\frac{θ_{1} - θ_{2}}{2})} =____(\cos (\frac{θ_{1} - θ_{2}}{2}) \neq 0)$

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Detailed Solution

Equation of any chord joining the points $P (θ_{1}) and P (θ_{2})$ to $\frac{x^{2}}{2023} - \frac{y^{2}}{2023} = 1$ is $\frac{x}{\sqrt{2023}} \cos (\frac{θ_{1} - θ_{2}}{2}) - \frac{y}{\sqrt{2023}} \sin (\frac{θ_{1} + θ_{2}}{2}) = \cos (\frac{θ_{1} + θ_{2}}{2})$
If it passes through focus $(\pm \sqrt{4046}, 0)$ , then we get $\pm \sqrt{2} \cos (\frac{θ_{1} - θ_{2}}{2}) = \cos (\frac{θ_{1} + θ_{2}}{2})$