If the line joining the points A(α) and B(β) on the ellipse x225+y29=1 is a focal chord, then one possible value of cot⁡α2cot⁡β2 is

Equation of given ellipse is x225+y29=1⇒a=5,b=3. Eccentricity, e=25−95=45Equation of chord joining A(α) and B(β) is xacos⁡α+β2+ybsin⁡α+β2=cos⁡α−β2→(1)(1) is a focal chord ⇒(1) passes through the focus ae, 0⇒aeacos⁡α+β2=cos⁡α−β2⇒cos⁡α2+β2cos⁡α2−β2=1e⇒cos⁡α2+β2+cos⁡α2−β2cos⁡α2+β2−cos⁡α2−β2=1+e1−e⇒2cos⁡α2cos⁡β2−2sin⁡α2sin⁡β2=1+e1−e⇒cot⁡α2cot⁡β2=e+1e−1=45+145−1=−9.

If the line joining the points A(α) and B(β) on the ellipse x225+y29=1 is a focal chord, then one possible value of cot⁡α2cot⁡β2 is

Equation of given ellipse is x225+y29=1⇒a=5,b=3. Eccentricity, e=25−95=45Equation of chord joining A(α) and B(β) is xacos⁡α+β2+ybsin⁡α+β2=cos⁡α−β2→(1)(1) is a focal chord ⇒(1) passes through the focus ae, 0⇒aeacos⁡α+β2=cos⁡α−β2⇒cos⁡α2+β2cos⁡α2−β2=1e⇒cos⁡α2+β2+cos⁡α2−β2cos⁡α2+β2−cos⁡α2−β2=1+e1−e⇒2cos⁡α2cos⁡β2−2sin⁡α2sin⁡β2=1+e1−e⇒cot⁡α2cot⁡β2=e+1e−1=45+145−1=−9.

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If the line joining the points $A (α) and B (β)$ on the ellipse $\frac{x^{2}}{25} + \frac{y^{2}}{9} = 1$ is a focal chord, then one possible value of $\cot \frac{α}{2} \cot \frac{β}{2}$ is

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Detailed Solution

Equation of given ellipse is $\frac{x^{2}}{25} + \frac{y^{2}}{9} = 1 \Rightarrow a = 5, b = 3 .$ Eccentricity, $e = \frac{\sqrt{25 - 9}}{5} = \frac{4}{5}$

Equation of chord joining $A (α) and B (β)$ is $\frac{x}{a} \cos (\frac{α + β}{2}) + \frac{y}{b} \sin (\frac{α + β}{2}) = \cos (\frac{α - β}{2}) \to (1)$

(1) is a focal chord $\Rightarrow (1)$ passes through the focus $(a e, 0)$

$\Rightarrow \frac{a e}{a} \cos (\frac{α + β}{2}) = \cos (\frac{α - β}{2}) \Rightarrow \frac{\cos (\frac{α}{2} + \frac{β}{2})}{\cos (\frac{α}{2} - \frac{β}{2})} = \frac{1}{e} \Rightarrow \frac{\cos (\frac{α}{2} + \frac{β}{2}) + \cos (\frac{α}{2} - \frac{β}{2})}{\cos (\frac{α}{2} + \frac{β}{2}) - \cos (\frac{α}{2} - \frac{β}{2})} = \frac{1 + e}{1 - e}$

$\Rightarrow \frac{2 \cos \frac{α}{2} \cos \frac{β}{2}}{- 2 \sin \frac{α}{2} \sin \frac{β}{2}} = \frac{1 + e}{1 - e} \Rightarrow \cot (\frac{α}{2}) \cot (\frac{β}{2}) = \frac{e + 1}{e - 1} = \frac{\frac{4}{5} + 1}{\frac{4}{5} - 1} = - 9 .$