120631 Let \(S \equiv \frac{x^2}{a^2}+\frac{y^2}{b^2}-1=0, S^{\prime} \equiv \frac{x^2}{\alpha^2}+\frac{y^2}{\beta^2}-1=0\) be two intersecting ellipses. If \(P(\operatorname{acos} \theta, b \sin \theta)\) and \(Q\left(\operatorname{acos}\left(\frac{\pi}{2}+\theta\right), b \sin \left(\frac{\pi}{2}+\theta\right)\right)\) are their points of intersection then \(\frac{1}{2}\left(a^2 \beta^2+b^2 \alpha^2\right)=\)

120632 Let \(P\left(\frac{\pi}{4}\right), Q\left(\frac{5 \pi}{4}\right), R\left(\frac{3 \pi}{4}\right), T\left(\frac{7 \pi}{4}\right)\) be the points on the hyperbola \(x^2-4 y^2-4=0\) in the parametric form. Then the area of the quadrilateral PQRT is (in square units)

120633 The end points of the major axis of an ellipse are \((2,4)\) and \((2,-8)\). If the distance between foci of this ellipse is 4 , then the equation of the ellipse is

120634 If \(x=h+a \sec \theta\) and \(y=k+b \operatorname{cosec} \theta\). Then

120635 The parametric representation of a point on the ellipse whose foci are \((3,0)\) and \((-1,0)\) and eccentricity \(2 / 3\), is

120631 Let \(S \equiv \frac{x^2}{a^2}+\frac{y^2}{b^2}-1=0, S^{\prime} \equiv \frac{x^2}{\alpha^2}+\frac{y^2}{\beta^2}-1=0\) be two intersecting ellipses. If \(P(\operatorname{acos} \theta, b \sin \theta)\) and \(Q\left(\operatorname{acos}\left(\frac{\pi}{2}+\theta\right), b \sin \left(\frac{\pi}{2}+\theta\right)\right)\) are their points of intersection then \(\frac{1}{2}\left(a^2 \beta^2+b^2 \alpha^2\right)=\)

120632 Let \(P\left(\frac{\pi}{4}\right), Q\left(\frac{5 \pi}{4}\right), R\left(\frac{3 \pi}{4}\right), T\left(\frac{7 \pi}{4}\right)\) be the points on the hyperbola \(x^2-4 y^2-4=0\) in the parametric form. Then the area of the quadrilateral PQRT is (in square units)

120633 The end points of the major axis of an ellipse are \((2,4)\) and \((2,-8)\). If the distance between foci of this ellipse is 4 , then the equation of the ellipse is

120634 If \(x=h+a \sec \theta\) and \(y=k+b \operatorname{cosec} \theta\). Then

120635 The parametric representation of a point on the ellipse whose foci are \((3,0)\) and \((-1,0)\) and eccentricity \(2 / 3\), is

120631 Let \(S \equiv \frac{x^2}{a^2}+\frac{y^2}{b^2}-1=0, S^{\prime} \equiv \frac{x^2}{\alpha^2}+\frac{y^2}{\beta^2}-1=0\) be two intersecting ellipses. If \(P(\operatorname{acos} \theta, b \sin \theta)\) and \(Q\left(\operatorname{acos}\left(\frac{\pi}{2}+\theta\right), b \sin \left(\frac{\pi}{2}+\theta\right)\right)\) are their points of intersection then \(\frac{1}{2}\left(a^2 \beta^2+b^2 \alpha^2\right)=\)

120632 Let \(P\left(\frac{\pi}{4}\right), Q\left(\frac{5 \pi}{4}\right), R\left(\frac{3 \pi}{4}\right), T\left(\frac{7 \pi}{4}\right)\) be the points on the hyperbola \(x^2-4 y^2-4=0\) in the parametric form. Then the area of the quadrilateral PQRT is (in square units)

120633 The end points of the major axis of an ellipse are \((2,4)\) and \((2,-8)\). If the distance between foci of this ellipse is 4 , then the equation of the ellipse is

120634 If \(x=h+a \sec \theta\) and \(y=k+b \operatorname{cosec} \theta\). Then

120635 The parametric representation of a point on the ellipse whose foci are \((3,0)\) and \((-1,0)\) and eccentricity \(2 / 3\), is

120631 Let \(S \equiv \frac{x^2}{a^2}+\frac{y^2}{b^2}-1=0, S^{\prime} \equiv \frac{x^2}{\alpha^2}+\frac{y^2}{\beta^2}-1=0\) be two intersecting ellipses. If \(P(\operatorname{acos} \theta, b \sin \theta)\) and \(Q\left(\operatorname{acos}\left(\frac{\pi}{2}+\theta\right), b \sin \left(\frac{\pi}{2}+\theta\right)\right)\) are their points of intersection then \(\frac{1}{2}\left(a^2 \beta^2+b^2 \alpha^2\right)=\)

120632 Let \(P\left(\frac{\pi}{4}\right), Q\left(\frac{5 \pi}{4}\right), R\left(\frac{3 \pi}{4}\right), T\left(\frac{7 \pi}{4}\right)\) be the points on the hyperbola \(x^2-4 y^2-4=0\) in the parametric form. Then the area of the quadrilateral PQRT is (in square units)

120633 The end points of the major axis of an ellipse are \((2,4)\) and \((2,-8)\). If the distance between foci of this ellipse is 4 , then the equation of the ellipse is

120634 If \(x=h+a \sec \theta\) and \(y=k+b \operatorname{cosec} \theta\). Then

120635 The parametric representation of a point on the ellipse whose foci are \((3,0)\) and \((-1,0)\) and eccentricity \(2 / 3\), is

120631 Let \(S \equiv \frac{x^2}{a^2}+\frac{y^2}{b^2}-1=0, S^{\prime} \equiv \frac{x^2}{\alpha^2}+\frac{y^2}{\beta^2}-1=0\) be two intersecting ellipses. If \(P(\operatorname{acos} \theta, b \sin \theta)\) and \(Q\left(\operatorname{acos}\left(\frac{\pi}{2}+\theta\right), b \sin \left(\frac{\pi}{2}+\theta\right)\right)\) are their points of intersection then \(\frac{1}{2}\left(a^2 \beta^2+b^2 \alpha^2\right)=\)

120632 Let \(P\left(\frac{\pi}{4}\right), Q\left(\frac{5 \pi}{4}\right), R\left(\frac{3 \pi}{4}\right), T\left(\frac{7 \pi}{4}\right)\) be the points on the hyperbola \(x^2-4 y^2-4=0\) in the parametric form. Then the area of the quadrilateral PQRT is (in square units)

120633 The end points of the major axis of an ellipse are \((2,4)\) and \((2,-8)\). If the distance between foci of this ellipse is 4 , then the equation of the ellipse is

120634 If \(x=h+a \sec \theta\) and \(y=k+b \operatorname{cosec} \theta\). Then

120635 The parametric representation of a point on the ellipse whose foci are \((3,0)\) and \((-1,0)\) and eccentricity \(2 / 3\), is