Parametric Representation: Ellipse

Q. The product of the perpendicular distance from the foci to any tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is

1. 
2. 
3. 
4. 

Q. If the tangents drawn at the points $O(0,0)$ and $P(1+\sqrt{5},2)$ on the circle $x^2+y^2-2x-4y=0$ intersect at the point $Q$, then the area of the triangle $OPQ$ is equal to

Q. The maximum length of chord of the ellipse $\frac{x^2}{8}+\frac{y^2}{4}=1$ such that eccentric angles of its extremities differ by $\frac{\pi }{2},$ is

Q. Let $P,Q,R$ be the points on the auxiliary circle of ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$ , such that $PQR$ is an equilateral triangle and $P^{\prime }{Q}^{\prime }{R}^{\prime }$ is corresponding triangle inscribed within the ellipse. Then centroid of the triangle $P^{\prime }{Q}^{\prime }{R}^{\prime }$ lies at

1. centre of the ellipse
2. focus of the ellipse
3. between one extremity of minor axis and centre of the ellipse
4. between focus and centre of the ellipse

Q. The equation of the lines through the points (2, 3) and making an intercept of length 2 unit between the lines y + 2x = 3 and y + 2x = 5, are

1. None of the above
2. x + 3 = 0, 3x + 4y = 12
3. y – 2 = 0, 4x – 3y = 6
4. x – 2 = 0, 3x + 4y = 18

Q. The area of the triangle formed by the asymptotes and any tangent to the hyperbola $x^2-y^2=a^2$

1. 
2. 
3. 
4. 

Q. The area of a quadrilateral ABCD where A = $(0,4,1),B=(2,3,–1),C=(4,5,0)$ and $D=(2,6,2)$ is equal to :

1. 9 sq. units
2. 18 sq. units
3. 27 sq. units
4. 81 sq. units

Q. Let $P,D$ be two points on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, whose eccentric angles differ by $\frac{\pi }{2}$. Then the locus of mid point of chord $PD$ is

1. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=2$
2. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{1}{2}$
3. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=4$
4. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{1}{4}$

Q. The ratio of the area enclosed by the locus of mid-point of PS and area of the ellipse where P is any point on the ellipse and S is the focus of the ellipse, is

1. 
2. 
3. 
4. 

Q. $P\left(\theta \right)$ and $Q(\theta +\frac{\pi }{2})$ are two points on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1.$ The locus of midpoint of the chord PQ is

1. 
2. 
3. 
4. 

Q. If a chord joining two points $A,B$ whose eccentric angles are $\alpha ,\beta$ cuts the major axis of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ at a distance $1$ from the centre, then $\left|3\tan \frac{\alpha }{2}\tan \frac{\beta }{2}\right|=$

Q. The line passing through the extremity $A$ of the major axis and extremity $B$ of the minor axis of the ellipse $x^2+9y^2=9$ meets its auxiliary circle at the point $M$. Then the area of the triangle with vertices at $A,M$ and the origin $O$ is

1. $\frac{29}{10}$ sq. units
2. $\frac{31}{10}$ sq. units
3. $\frac{21}{10}$ sq. units
4. $\frac{27}{10}$ sq. units

Q.

Let $P$ be a variable point on the ellipse $\frac{x^2}{100}+\frac{y^2}{64}=1$ with foci $F_1$ and $F_2$. If $A$ is the area of triangle $P{F}_1{F}_2$, then the maximum possible value of $A$ is

1. $120$
2. $48$
3. $152$
4. $24$

Q. The equation of the curve whose parametric equations are $x=1+4\cos \theta ,y=2+3\sin \theta ,\theta \in R,$ is

1. $9x^2+16y^2-18x-64y-71=0$
2. $9x^2+16y^2-18x-64y+71=0$
3. $16x^2+9y^2-64x-18y-71=0$
4. $16x^2+9y^2-64x-18y+71=0$

Q. Let $P$ be a variable point on the ellipse $\frac{x^2}{100}+\frac{y^2}{64}=1$ with foci $F_1$ and $F_2.$ If $A$ is the area of triangle $P{F}_1{F}_2,$ then the maximum possible value of $A$ is

Q. If a curve is represented parametrically by the equations
$x=\sin (t+\frac{7\pi }{12})+\sin (t-\frac{\pi }{12})+\sin (t+\frac{3\pi }{12})$
$y=\cos (t+\frac{7\pi }{12})+\cos (t-\frac{\pi }{12})+\cos (t+\frac{3\pi }{12}),$
then the value of $\frac{d}{dt}(\frac{x}{y}-\frac{y}{x})$ at $t=\frac{\pi }{8}$ is

Q. If the tangent to $y^2=4ax$ at the point $(a{t}^2,2at)$ where $|t|>1$ is a normal to $x^2-y^2=a^2$ at the point $(a\sec \theta ,a\tan \theta ),$ then

1. $t=-\text{cosec}\theta$
2. $t=-\sec \theta$
3. $t=2\tan \theta$
4. $t=2\cot \theta$

Q.

The area of the circle and the area of the regular polygon of n sides and of perimeter equal to that of circle are in the ratio of

1. 

2. 

3. 

4. 

Q. If $\left(\sqrt{3}\right)bx+ay=2ab$ is tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ , then eccentric angle $\theta$ is

1. $\frac{\pi }{4}$
2. $\frac{\pi }{6}$
3. $\frac{\pi }{2}$
4. $\frac{\pi }{3}$

Q.

Let $P$ be a variable point on the ellipse $\frac{x^2}{100}+\frac{y^2}{64}=1$ with foci $F_1$ and $F_2$. If $A$ is the area of triangle $P{F}_1{F}_2$, then the maximum possible value of $A$ is

1. $120$
2. $48$
3. $152$
4. $24$

Q. The equation with the parametric equations x = 1 + cos $\theta$, y = 2 + 3 sin $\theta$ represents an ellipse.

1. False
2. True

Q. Let ` $P$' be a variable point on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with foci $S(ae,0)$ and $S^{\prime }(-ae,0)$ . lf $A$ is the area of the triangle $PS{S}^1$, then the maximum value of $A$ (where $e$ is eccentricity and $b^2=a^2(1-e^2))$ is

1. $\frac{ab}{2}$
2. 2 abe
3. abe
4. 4abe

Q. The maximum length of chord of the ellipse $\frac{x^2}{8}+\frac{y^2}{4}=1$ such that eccentric angles of its extremities differ by $\frac{\pi }{2},$ is

Q.

If $\cos \alpha =\frac{2}{3},$ then the range of values of $\varphi$ on the ellipse
$x^2+4y^2=4$ falls inside the circle $x^2+y^2+4x+3=0$ is

1. $-\alpha ,\alpha$

2. $0,\alpha$

3. $\alpha ,\pi$

4. $\pi -\alpha ,\pi +\alpha$

Q. A plane $x+y+z=1$ intersects with a sphere which passes through the origin. The image of the center of the sphere about the plane lies on the line $6x+8=6y+8=-3z-4$. If the area of the cross section is $16\pi$, then

1. Radius of the sphere is $\sqrt{\frac{73}{3}}$
2. Coordinates of the center is $(2+\sqrt{\frac{37}{18}},2+\sqrt{\frac{37}{18}},2-\sqrt{\frac{74}{9}})$
3. Coordinates of the center is $(2-\sqrt{\frac{37}{18}},2-\sqrt{\frac{37}{18}},2+\sqrt{\frac{74}{9}})$
4. Radius of the sphere is $2\sqrt{\frac{73}{3}}$

Q. If $\left(\sqrt{3}\right)bx+ay=2ab$ is tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ , then eccentric angle $\theta$ is

1. 
2. 
3. 
4. 

Q. Assertion :Bragg's equation has no solution, if $n=2$ and $\lambda >d$ Reason: Bragg's equation is $n\lambda =2d\sin \theta$

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Both Assertion and Reason are incorrect

Q. The area of the triangle formed by the axes and the line $(\cosh \alpha -sinh\alpha )x+(\cosh \alpha +sinh\alpha )y=2$ in square units is :

1. $4$
2. $3$
3. $1$
4. $2$

Q. The ratio of the area enclosed by the locus of mid-point of PS and area of the ellipse where P is any point on the ellipse and S is the focus of the ellipse, is

1. $\frac{1}{2}$
2. $\frac{1}{3}$
3. $\frac{1}{5}$
4. $\frac{1}{4}$

Q.

If $\cos \alpha =\frac{2}{3},$ then the range of values of $\varphi$ on the ellipse
$x^2+4y^2=4$ falls inside the circle $x^2+y^2+4x+3=0$ is

1. $-\alpha ,\alpha$

2. $0,\alpha$

3. $\alpha ,\pi$

4. $\pi -\alpha ,\pi +\alpha$