An ellipse frac{(x-x_0)^2}{a^2} + frac{(y-y_0 | vedclass

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(C) Since the ellipse is tangent to both axes in the first quadrant,its center is $(a, b)$. The foci are $(a \pm ae, b)$ or $(a, b \pm ae)$. Given the

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$\frac{1}{2}$

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(C) Since the ellipse is tangent to both axes in the first quadrant,its center is $(a, b)$. The foci are $(a \pm ae, b)$ or $(a, b \pm ae)$. Given the geometry,the center is $(a, b)$ and the distance from the origin to the center is $\sqrt{a^2 + b^2}$.For an ellipse tangent to the axes,$x_0 = a$ and $y_0 = b$. The foci are $(a \pm ae, b)$.Given $	riangle OF_1F_2$ is isosceles with $\angle OF_1F_2 = 120^{\circ}$,we have $OF_1 = F_1F_2 = 2ae$.Using the law of cosines in $	riangle OF_1F_2$ or coordinate geometry properties,we find $e = \frac{1}{2}$.

List-$I$	List-$II$
$(P)$ Directrix corresponding to the focus $(-3, 0)$	$(1)$ $y = 4$
$(Q)$ Tangent at the vertex $(0, 4)$	$(2)$ $3x = 25$
$(R)$ Latus rectum through $(3, 0)$	$(3)$ $x = 3$
	$(4)$ $y + 4 = 0$
	$(5)$ $x + 3 = 0$
	$(6)$ $3x + 25 = 0$

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