Let S = x , y ∈ R 2 : y 2 1 + r - x 2 1 - r = 1 , where r ≠ ± 1 . Then S represents:

An ellipse whose eccentricity is 2 r + 1 , when r > 1

An ellipse whose eccentricity is 1 r + 1 , when r > 1 .

A hyperbola whose eccentricity is 2 r + 1 , when 0 < r < 1 .

A hyperbola whose eccentricity is 2 1 - r , when 0 < r < 1

Consider the point P43 and x216y291 Which of the following is not true

MEDIUM

Mathematics>Geometry - II>Expressing Geometric Properties with Equations>Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant

The locus of the point of intersection of the straight lines,

t x - 2 y - 3 t = 0

and

x - 2 t y + 3 = 0 (t \in R),

is:

HARD

Mathematics>Geometry - II>Expressing Geometric Properties with Equations>Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant

Equation

\frac{1}{r} = \frac{1}{8} + \frac{3}{8} \cos θ

represents

MEDIUM

Mathematics>Geometry - II>Expressing Geometric Properties with Equations>Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant

Which of the following is the equation of a hyperbola?

MEDIUM

Mathematics>Geometry - II>Expressing Geometric Properties with Equations>Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant

Let

a

and

b

respectively be the semi-transverse and semi-conjugate axes of a standard hyperbola whose eccentricity satisfies the equation

9 e^{2} - 18 e + 5 = 0

. If

S (5, 0)

is a focus and

5 x = 9

is the corresponding directrix of this hyperbola, then

a^{2} - b^{2}

is equal to

EASY

Mathematics>Geometry - II>Expressing Geometric Properties with Equations>Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant

If the eccentricity of the hyperbola

x^{2} - y^{2} {cosec}^{2} α = 25

is

\sqrt{5}

times the eccentricity of the ellipse

x^{2} {cosec}^{2} α + y^{2} = 5,

then

α

is equal to

MEDIUM

Mathematics>Geometry - II>Expressing Geometric Properties with Equations>Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant

A hyperbola whose transverse axis is along the major axis of the conic

\frac{x^{2}}{3} + \frac{y^{2}}{4} = 4

and has vertices at the foci of the conic. If the eccentricity of the hyperbola is

\frac{3}{2}

, then which of the following points does not lie on the hyperbola

?

HARD

Mathematics>Geometry - II>Expressing Geometric Properties with Equations>Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant

The foci of the hyperbola

16 x^{2} - 9 y^{2} - 64 x + 18 y - 90 = 0

are

MEDIUM

Mathematics>Geometry - II>Expressing Geometric Properties with Equations>Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant

The eccentricity of the hyperbola whose length of the latus rectum is equal to

8

and the length of its conjugate axis is equal to half of the distance between its foci, is

EASY

Mathematics>Geometry - II>Expressing Geometric Properties with Equations>Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant

If the vertices of a hyperbola be at

(- 2, 0)

and

(2, 0)

and one of its foci be at

(- 3, 0)

, then which one of the following points does not lie on this hyperbola ?

MEDIUM

Mathematics>Geometry - II>Expressing Geometric Properties with Equations>Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant

Let

S = \{(x, y) \in R^{2} : \frac{y^{2}}{1 + r} - \frac{x^{2}}{1 - r} = 1\},

where

r \neq \pm 1 .

Then

S

represents:

EASY

Mathematics>Geometry - II>Expressing Geometric Properties with Equations>Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant

If

5 x + 9 = 0

is the directrix of the hyperbola

16 x^{2} - 9 y^{2} = 144,

then its corresponding focus is:

HARD

Mathematics>Geometry - II>Expressing Geometric Properties with Equations>Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant

Let

0 < θ < \frac{π}{2} .

If the eccentricity of the hyperbola

\frac{x^{2}}{\cos^{2} θ} - \frac{y^{2}}{\sin^{2} θ} = 1

is greater than

2,

then the length of its latus rectum lies in the interval:

MEDIUM

Mathematics>Geometry - II>Expressing Geometric Properties with Equations>Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant

If a directrix of a hyperbola centered at the origin and passing through the point

(4, - 2 \sqrt{3})

is

5 x = 4 \sqrt{5}

and its eccentricity is

e

, then:

MEDIUM

Mathematics>Geometry - II>Expressing Geometric Properties with Equations>Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant

The eccentricity of the hyperbola whose length of its conjugate axis is equal to half of the distance between its foci, is

EASY

Mathematics>Geometry - II>Expressing Geometric Properties with Equations>Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant

The latus rectum of the hyperbola

3 x^{2} - 2 y^{2} = 6

is

MEDIUM

Mathematics>Geometry - II>Expressing Geometric Properties with Equations>Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant

If equation

(10 x - 5)^{2} + (10 y - 4)^{2} = λ^{2} (3 x + 4 y - 1)^{2}

represents a hyperbola, then

MEDIUM

Mathematics>Geometry - II>Expressing Geometric Properties with Equations>Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant

If

e_{1}

and

e_{2}

are the eccentricities of a hyperbola

3 x^{2} - 3 y^{2} = 25

and its conjugate, then

EASY

Mathematics>Geometry - II>Expressing Geometric Properties with Equations>Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant

The vertices of the hyperbola are at

(- 5, - 3)

and

(- 5, - 1)

and the extremities of the conjugate axis are at

(- 7, - 2)

and

(- 3, - 2),

then the equation of the hyperbola is

EASY

Mathematics>Geometry - II>Expressing Geometric Properties with Equations>Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant

A hyperbola has its centre at the origin, passes through the point

(4, 2)

and has transverse axis of length

4

along the

x - a x i s .

Then the eccentricity of the hyperbola is:

EASY

Mathematics>Geometry - II>Expressing Geometric Properties with Equations>Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant

The equation of the hyperbola with vertices

(3, 0), (- 3, 0)

and semi-latus rectum

4

is given by:

Consider the point $P (4, 3)$ and $\frac{x^{2}}{16} - \frac{y^{2}}{9} = 1$ . Which of the following is not true?

Important Questions on Expressing Geometric Properties with Equations

Learn and Prepare for any exam you want !

Consider the point P4,3 and x216-y29=1. Which of the following is not true?

Important Questions on Expressing Geometric Properties with Equations

Learn and Prepare for any exam you want !

Consider the point $P (4, 3)$ and $\frac{x^{2}}{16} - \frac{y^{2}}{9} = 1$ . Which of the following is not true?