Parametric Representation: Hyperbola

Q. Let $C$ be a conic whose parametric equations are $x=\frac{e^t+e^{-t}}{2}$ and $y=\frac{e^t-e^{-t}}{3};$ where $t$ is a parameter. Then which of the following is/are true?

1. $C$ is a hyperbola with eccentricity $\frac{\sqrt{13}}{3}$
2. $C$ is a hyperbola with eccentricity $\frac{\sqrt{5}}{3}$
3. $C$ intersects $y-$ axis at two points
4. Auxilary circle of $C$ passes through the point $(\frac{1}{2},-\frac{\sqrt{3}}{2})$

Q.

The parametric form of the ellipse $4{(x+1)}^2+{(y-1)}^2=4$ is

1. $x=\cos \theta –1,y=2\sin \theta –1$

2. $x=2\cos \theta –1,y=\sin \theta +1$

3. $x=\cos \theta –1,y=2\sin \theta +1$

4. $x=\cos \theta +1,y=2\sin \theta +1$

5. $x=\cos \theta +1,y=2\sin \theta –1$

Q.

The normal to the rectangular hyperbola $xy=c^2$ at the point '$t_1$' meets the curve again at the point '$t_2$'. Then the value of $t_1^3{t}_2$is

1. 1

2. -1

3. c

4. -c

Q.

The angle between two diagonals of a cube will be
[MP PET 1996, 2000; RPET 2000, 02; UPSEAT 2004]

1. 
   

2. Variable

3. None of these

4. 
   

Q. If a triangle is inscribed in a rectangular hyperbola, the shortest distance of the orthocentre to the hyperbola is

Q. A derivable function $f:R^{+}\to R$ satisfies the condition $f\left(x\right)-f\left(y\right)\ge \ln \frac{x}{y}+x-y;\forall x,y\in R^{+}$. If $g$ denotes the derivative of $f$ then the value of the sum ${\sum }_{n=1}^{100}g\left(\frac{1}{n}\right)$ is $1030k$.Find the value of $k$.

Q.

If a triangle is inscribed in a rectangular hyperbola, its orthocentre lies

1. inside the curve

2. Outside the curve

3. on the curve

4. none of these

Q. The slope of the side BC of a rectangle ABCD is $\frac{2}{3}$. Find:
(i) The slope of the side AB.

Q.

The parametric form the curve
$\frac{(x+1{)}^2}{16}-\frac{(y-2{)}^2}{4}=1$ is

1. $(4\sec \theta ,2\tan \theta )$

2. $(4\sec \theta +1,2\tan \theta -2)$

3. $(4\sec \theta -1,2\tan \theta +2)$

4. $(4\sec \theta -4,2\tan \theta -2)$

Q. Let the circle $x^2+y^2-8x=0$ and hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$ intersect at the points $A$ and $B$, where $A$ is in first quadrant. If an ellipse is constructed with vertices at $A,B$ and having eccentricity $\frac{1}{\sqrt{3}}$, then the distance between its foci is

Q. The curve described parametrically by $y=B\tan \theta$ and $x=A\sec \theta$ represents

1. Ellipse
2. Parabola
3. Hyperbola
4. Circle

Q. If $PQ$ is a double ordinate of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ such that $OPQ$ is an equilateral triangle, $O$ being the centre of the hyperbola, then
(where $e$ is the eccentricity of the hyperbola)

1. $1<e<\frac{2}{\sqrt{3}}$
2. $1<e<\frac{3}{2}$
3. $e=\frac{2}{\sqrt{3}}$
4. $e>\frac{2}{\sqrt{3}}$

Q. The curve passes through $(1,1)$ satisfies the differential equation
$\frac{dy}{dx}+\sqrt{(1-\frac{1}{x^2})(1-\frac{1}{y^2})}=0$
If it passes through $(\sqrt{2},k)$ then the possible value(s) of $\left[k\right]$ is/are
(where [.] denote the greatest integer function)

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

Q. The equation of the parabola whose vertex and focus lie on the axis of $x$ at distances $a$ and $a_1$ from the origin respectively is:

1. $y^2=4(a_1-a)(x-a_1)$
2. $y^2=4(a_1-a)(x-a)$
3. none of these
4. $y^2=4(a_1-a)x$

Q.

The normal to the rectangular hyperbola $xy=c^2$ at the point '$t_1$' meets the curve again at the point '$t_2$'. Then the value of $t_1^3{t}_2$is

1. 1

2. -1

3. c

4. -c

Q. If one vertex of an equilateral triangle of side ${}^{\prime }{a}^{\prime }$ lies at the origin and the other lies on the line $x-\sqrt{3}y=0$, then the co-ordinates of the third vertex are :

1. $(0,a)$
2. $(\frac{\sqrt{3}a}{2},-\frac{a}{2})$
3. $(0,-a)$
4. $(-\frac{\sqrt{3}a}{2},\frac{a}{2})$

Q. The coefficient of linear expansion of cyrstal in one direction is ${\alpha }_1$ and that in every direction perpendicular to it ${\alpha }_2$ . the coefficient of cubical expansion is :

1. $2{\alpha }_1+2{\alpha }_2$
2. ${\alpha }_1+{\alpha }_2$
3. ${\alpha }_1+2{\alpha }_2$
4. none of these

Q. Let $C$ be a conic whose parametric equations are $x=\frac{e^t+e^{-t}}{2}$ and $y=\frac{e^t-e^{-t}}{3};$ where $t$ is a parameter. Then which of the following is/are true?

1. $C$ is a hyperbola with eccentricity $\frac{\sqrt{13}}{3}$
2. $C$ is a hyperbola with eccentricity $\frac{\sqrt{5}}{3}$
3. $C$ intersects $y-$ axis at two points
4. Auxilary circle of $C$ passes through the point $(\frac{1}{2},-\frac{\sqrt{3}}{2})$

Q. If the centroid of an equilateral triangle is $(1,1)$ and its one vertex is $(-1,2)$ then the equation of its circumcircle is

1. $x^2+y^2-2x-2y-3=0$
2. $x^2+y^2+2x+2y-3=0$
3. $x^2+y^2+2x-2y-3=0$
4. none of these

Q. If $A,B$ and $C$ are interior angles of a triangle $ABC$, then $\sin \left(\frac{B+C}{2}\right)=\cos \frac{A}{2}$.

1. True
2. False

Q.

What is the equation of a curve given by the parametric form $x=9+6sec\theta ;y=-2-4tan\theta$.

1. $\frac{(x+9{)}^2}{36}-\frac{(y-2{)}^2}{16}=1$

2. $\frac{(x-9{)}^2}{36}-\frac{(y+2{)}^2}{16}=1$

3. $\frac{(x-6{)}^2}{81}-\frac{(y+4{)}^2}{4}=1$

4. $\frac{(x+6{)}^2}{81}-\frac{(y-4{)}^2}{4}=1$

Q. '$O$' is the vertex of the parabola $y^2=8x$ and L is the upper end of the latus rectum. If LH is drawn perpendicular to $OL$ meeting $OX$ in $H$, then the length of the double ordinate through $H$ is $\lambda \sqrt{5}$ where $\lambda$ is equal to

1. $2$
2. $4$
3. $6$
4. $8$

Q. If $3x^2+xy-4y^2=0$ is the equation of a pair of conjugate diameters of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ then its eccentricity is

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

Q.

A conic section is defined by the equations x = - 1 + sect y = 2 + z tant. The coordinates of the foci are,

1. $(-1-\sqrt{10},2)and(-1+\sqrt{10},2)$

2. $(-1-\sqrt{8},2)and(-1+\sqrt{8},2)$

3. $(-1,2-\sqrt{8})and(-1,2+\sqrt{8})$

4. $(-1,2-\sqrt{10})and(-1,2+\sqrt{10})$

Q. If two vertices of an equilateral triangle have integral coordinates, then the third vertex will have:

1. integral coordinates
2. coordinates which are rational
3. at least one coordinate irrational
4. coordinates which are irrational

Q.

The normal to the rectangular hyperbola $xy=c^2$ at the point '$t_1$' meets the curve again at the point '$t_2$'. Then the value of $t_1^3{t}_2$is

1. 1

2. -1

3. c

4. -c

Q.

In parametric representation of a standard hyperbola in $\theta$ terms, if $x=asec\theta$. What is y.

1. 

2. 

3. 

4. 

Q.

A conic section is defined by the equations x = - 1 + sect y = 2 + z tant. The coordinates of the foci are,

1. ~and~ (-1, 2+\sqrt{10})\)

2. ~and~ (-1+\sqrt{8}, 2)\)

3. ~and~ (-1, 2+\sqrt{8})\)

4. ~and~ (-1+\sqrt{10}, 2)\)

Q. If $A+B={90}^{}$, $secA=a,cotB=b$ then " $a^2-b^2=0$. "if this is right then true and otherwise false ?

1. True
2. False

Q. Show that the curves $xy=a^2$ and $x^2+y^2=2a^2$ touch each other.