sqrt{1-\frac{b^2}{a^2}} \\ & =\sqrt{1-\frac{75}{100}} \\ & e_E=\frac{1}{2} \end{aligned}$$

$$\therefore e_H=2$$ [ as eccentricity of hyperbola is reciprocal of eccentricity of ellipse]

Transverse axis of hyperbola $$=\alpha$$

Conjugate axis of hyperbola $$=\beta$$

Also, foci of ellipse $$(1 \pm a e, 1)$$

$$\begin{aligned} & =\left(1 \pm\left(10 \times \frac{1}{2}\right), 1\right) \\ & =(1 \pm 5,1) \\ & =(6,1) \text { and }(-4,1) \end{aligned}$$

Distance between foci $$=10$$

$$\begin{aligned} & 2 a e=10 \\ & \Rightarrow a=\frac{5}{2} \end{aligned}$$

$$\begin{aligned} & \text { also, } e^2=1+\frac{b^2}{a^2} \\ & \begin{aligned} 4 & =1+\frac{4 b^2}{25} \\ b^2 & =\frac{75}{4} \\ b & =\frac{\sqrt{75}}{2} \end{aligned} \\ & \Rightarrow \quad \alpha=5 \\ & \text { and } \beta=\sqrt{75} \\ & 3 \alpha^2+2 \beta^2=3(5)^2+2(75)=225 \end{aligned}$$

19:T57f,

$$E: \frac{(x-1)^2}{100}+\frac{(y-1)^2}{75}=1$$

$$\begin{aligned} \text { दीर्घवृत्त की उत्केंद्रता, } e_E & =\sqrt{1-\frac{b^2}{a^2}} \\ & =\sqrt{1-\frac{75}{100}} \\ & e_E=\frac{1}{2} \end{aligned}$$

$$\therefore e_H=2$$ [क्योंकि अतिपरवलय की उत्केंद्रता, दीर्घवृत्त की उत्केंद्रता का व्युत्क्रम है]

अतिपरवलय का अनुप्रस्थ अक्ष $$=\alpha$$

अतिपरवलय का संयुग्मी अक्ष $$=\beta$$

साथ ही, दीर्घवृत्त की नाभियाँ $$(1 \pm a e, 1)$$

$$\begin{aligned} & =\left(1 \pm\left(10 \times \frac{1}{2}\right), 1\right) \\ & =(1 \pm 5,1) \\ & =(6,1) \text { और }(-4,1) \end{aligned}$$

नाभियों के बीच की दूरी $$=10$$

$$\begin{aligned} & 2 a e=10 \\ & \Rightarrow a=\frac{5}{2} \end{aligned}$$

$$\begin{aligned} & \text { साथ

Hyperbola

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