General Equation of Hyperbola

Q.

The distance between the foci of a hyperbola is 16 and its ecentricity is $\sqrt{2},$ then equation of the hyperbola is

1. $x^2+y^2=16$

2. $x^2-y^2=32$

3. $x^2-y^2=16$

4. $x^2+y^2=32$

Q. An ellipse intersects the hyperbola $2x^2-2y^2=1$ orthogonally. The eccentricity of the ellipse is the reciprocal of that of hyperbola. If the axis of the ellipse are along the coordinate axis, then

1. equation of ellipse is $x^2+2y^2=4$
2. coordinates of foci of ellipse are $(\pm 1,0)$
3. equation of ellipse is $x^2+2y^2=2$
4. coordinates of foci of ellipse are $(\pm \sqrt{2},0)$

Q.

Equation of the ellipse whose axes are the axes of coordinates and which passes through the point $(-3,1)$ and has eccentricity $\sqrt{\frac{2}{5}}$ is

1. $5x^2+3y^2–48=0$

2. $3x^2+5y^2–15=0$

3. $5x^2+3y^2–32=0$

4. $3x^2+5y^2–32=0$

Q.

The equation of the hyperbola whose foci are (6, 4) and (-4, 4) and eccentricity 2. is

1. $\frac{(x-1{)}^2}{25/4}-\frac{(y-4{)}^2}{75/4}=1$

2. $\frac{(x+1{)}^2}{25/4}-\frac{(y+4{)}^2}{75/4}=1$

3. $\frac{(x-1{)}^2}{75/4}-\frac{(y-4{)}^2}{25/4}=1$

4. none of these

Q. If the latus rectum of a hyperbola forms an equilateral triangle with the vertex at the centre of the hyperbola, then eccentricity of the hyperbola is

1. 
2. 
3. 
4. 

Q.

The area of the triangle formed by the coordinate axes and a tangent to the curve $\mathrm{xy}={\mathrm{a}}^2$ at the point $\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)$ on it is

1. $\frac{{\mathrm{a}}^2{\mathrm{x}}_1}{{\mathrm{y}}_1}$

2. $\frac{{\mathrm{a}}^2{\mathrm{y}}_1}{{\mathrm{x}}_1}$

3. $2{\mathrm{a}}^2$

4. $4{\mathrm{a}}^2$

Q.

$\text{Find the centre, eccentricity, foci and directrices of the hyperbola}\left(i\right)16x^2-9y^2+32x+36y-164=0\left(ii\right)x^2-y^2+4x=0\left(iii\right)x^2-3y^2-2x=8$

Q. If P is a point which moves inside an equilateral triangle of side length ‘a’ such that it is nearer to any angular bisector of the triangle than to any of its sides, then the area of the region in which P lies is __________ sq units

1. 
2. 
3. 
4. 

Q. If the length of latus rectum of a hyperbola $\frac{x^2}{k}-\frac{y^2}{25}=-1$ is $\frac{22}{5}$ units, then its $e$ (eccentricity) is

1. $\frac{7}{3}$
2. $\frac{7}{5}$
3. $\frac{6}{5}$
4. $\frac{7}{2}$

Q.

$\text{Find the equation of the hyperbola, referred to its principal axes as axes of coordinates, in the following cases:}\left(i\right)\text{the distance between the foci =16 and eccentricity}=\sqrt{2}\left(ii\right)\text{conjugate axis is 5 and the distance between foci=13}\left(iii\right)\text{conjugate axis is 7 and passes through the point (3, -2)}$

Q. If two lines have direction cosines $l_1,m_1,n_1$ and $l_2,m_2,n_2$ then the condition for these lines being parallel to each other would be -

1. None of these
2. 
3. Both A and B
4. 

Q.

Let $e_1$ and $e_1$ be the eccentricities of the ellipse, $\left(\frac{x^2}{25}\right)+\left(\frac{y^2}{b^2}\right)=1\left(b<5\right)$ and the hyperbola, $\left(\frac{x^2}{16}\right)-\left(\frac{y^2}{b^2}\right)=1$ respectively satisfying $e_1{e}_2=1$. If $\alpha$ and $\beta$ are the distances between the foci of the ellipse and the foci of the hyperbola respectively, then the ordered pair $\left(\alpha ,\beta \right)$ is equal to:

1. $\left(8,12\right)$

2. $\left(\frac{24}{5},10\right)$

3. $\left(\frac{20}{3},12\right)$

4. $\left(8,10\right)$

Q. A line parallel to the straight line $2x-y=0$ is tangent to the hyperbola
$\frac{x^2}{4}-\frac{y^2}{2}=1$ at the point $(x_1,y_1)$. Then $x_1^2+5y_1^2$ is equal to :

1. $6$
2. $10$
3. $8$
4. $5$

Q. The equation of the hyperbola whose directrix is $x+2y=1$, focus $(2,1)$ and eccentricity $2$ is

1. $x^2-16xy-11y^2-12x+6y+21=0$
2. $3x^2+16x+15y^2-4x-14y-1=0$
3. $x^2+16x+11y^2-12x-6y+21=0$
4. $3x^2+16x+15y^2-4x-14y+1=0$

Q. For a parabola, if $L_1:x=y+1$ is the axis of symmetry, $L_2:x+y=5$ is tangent at vertex and $L_3:y=4$ is a tangent at a point $P,$ then which of the following is (are) CORRECT ?

1. Focus of the parabola is $(1,0)$
2. The equation of circumcircle of the triangle formed by the tangent and normal at point $P$ and axis of parabola is $x^2+y^2-2x=31$
3. The length of latus rectum of $y^2=8\sqrt{2}x$ and the given parabola is equal
4. The area of the quadrilateral formed by the tangents and normals at the extremities of the latus rectum of the given parabola is $64$ sq. units

Q.

Equation of the hyperbola whose vertices are $(\pm 3,0)$ and foci at $(\pm 5,0)$ is

1. $9x^2-25y^2=81$

2. $16x^2-9y^2=144$

3. $9x^2-16y^2=144$

4. $25x^2-9y^2=225$

Q.

$\text{A point moves in a plane so that its distances PA and PB from two fixed points A and B in the plane satisfy the relation PA-PB}=k(k\ne 0),thenthelocusofPis$

1. a hyperbola

2. a branch of the hyperbola

3. an ellipse

4. a parabola

Q. If foci of $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ concide with the foci of $\frac{x^2}{25}+\frac{y^2}{9}=1$ and eccentricity of the hyperbola is $2$, then

1. $a^2+b^2=16$
2. end points of latus rectum will be $(\pm 4,\pm 6)$
3. ratio of lengths of transverse axis to conjugate axis is $1:\sqrt{3}$
4. length of latus rectum of the hyperbola $=12$ unit

Q. The equation of the hyperbola whose vertices are $(\pm 6,0)$ and one of its directrix is $x=4$ is

1. $\frac{y^2}{25}-\frac{x^2}{36}=1$

2. $\frac{x^2}{45}-\frac{y^2}{36}=1$

3. $\frac{x^2}{36}-\frac{y^2}{45}=1$
4. $\frac{x^2}{25}-\frac{y^2}{29}=1$

Q.

The equation of the conic with focus at (1, -1) directrix x-y+1=0 and eccentricity$\sqrt{2}$ is

1. $x^2-y^2=1$

2. $2xy-4x+4y+1=0$

3. $xy=1$

4. $2xy+4y-4y-1=0$

Q. The equation of the transverse and conjugate axis of the hyperbola $16x^2-y^2+64x+4y+44=0$ are

1. $x+2=0,y+2=0$
2. $x=2,y+2=0$
3. $x=2,y=2$
4. $x+2=0,y=2$

Q.

In each of the following find the equation of the hyperbola satisfying the given conditions.:

$\left(i\right)vertices(\pm 2,0),foci(\pm 3,0)$

$\left(ii\right)vertices(0,\pm ,5),foci(0,\pm ,8)$

$\left(iii\right)vertices(0,\pm ,3),foci(0,\pm ,5)$

$\left(iv\right)vertices(\pm 5,0),transverseaxis=8$

$\left(v\right)foci(0,\pm 13),conjugateaxis=24$

$\left(vi\right)foci(\pm 3\sqrt{5}),thelatus-rectum=8$

$\left(vii\right)foci(\pm ,4,0),thelatus-rectum=12$

$\left(viii\right)vertices(0,\pm 6),e=\frac{5}{3}.$

$\left(ix\right)foci(0,\pm \sqrt{1}0),passingthrough(2,3)$

$\left(x\right)foci(0,\pm 12),latus-rectum=36$

Q.

The equation of the directrix of a hyperbola is x-y+3 =0.Its ficys us(-1, 1) and eccentricity 3.Find the equation of the hyperbola.

Q. The eccentricity of the hyperbola whose asymptotes are $3x+4y=10$ and $4x-3y=5$ is

1. $\sqrt{3}$
2. $\sqrt{2}$
3. $\sqrt{5}$
4. $\sqrt{6}$

Q. If the ellipse $4x^2+9y^2=36$ and the hyperbola $a^2{x}^2-y^2=4$ intersect at right angles, then which among the following options are correct

1. $a=2$
2. $a=-4$
3. Equation of circle through the points of intersection of two conic is $x^2+y^2=5$
4. Equation of circle through the points of intersection of two conic is $x^2+y^2=25$

Q. If the chord $x\cos \alpha +y\sin \alpha =p$ of the hyperbola $\frac{x^2}{16}-\frac{y^2}{18}=1$ subtends a right angle at the centre, and the diameter of the circle, concentric with the hyperbola to which the given chord is a tangent is $d$ units, then the value of $\frac{d}{4}$ is

Q.

The equation of the hyperbola whose centre is(6, 2) one focus is (4, 2) and of eccenticity 2 is

1. $(x-6{)}^2-2(y-2{)}^2=1$

2. $3(x-6{)}^2-(y-2{)}^2=3$

3. $(x-6{)}^2-3(y-2{)}^2=1$

4. $2(x-6{)}^2-(y-2{)}^2=3$

Q. Let $16x^2-3y^2-32x-12y=44$ represent a hyperbola. Then

1. length of the transverse axis is $2\sqrt{3}$
   
2. length of each latus rectum is $32/\sqrt{3}$
3. eccentricity is $\sqrt{19/3}$
4. equation of a directrix is $x=\frac{\sqrt{19}}{3}$

Q. Find the equation of the hyperbola, referred to its principal axes as axes of coordinates, in following cases :
Conjugate axis is $7$ and passes through the point $(3,-2)$

Q.

Find the equation of the hyperbola whose

(i) focus is at (5, 2), vertex at (4, 2) and centre at(3, 2)

(ii) focus is at (4, 2), centre at (6, 2) and e=2.