Equation of Ellipse MCQ Quiz in मल्याळम - Objective Question with Answer for Equation of Ellipse - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

Concept:         

Concept:

The standard Equation of Ellipse is 

Major Axis: x-axis

c² = a² - b² 

Coordinate of foci (± c, 0)

Vertices (±a, 0)

Major axis = 2a

Minor Axis = 2b

Eccentricity = e = 

Latus Rectum = 

Major Axis: y-axis

c2 = b2 - a2 

Coordinate of foci (0, ± c)

Vertices (0, ±b)

Major axis = 2b

Minor Axis = 2a

Eccentricity = e = 

Latus Rectum = 

Calculation:

Given:

Equation of the ellipse is 

Here, a² = 100 and b² = 400

a = 10 and b = 20

Here, y-axis is the major axis.

Statement I:  Vertices of the ellipse are (0, ± 20)

Vertices (0, ±b) = (0, ± 20)

Statement I is correct.

Statement II: Foci of the ellipse are (0. ±10√3)

c2 = b2 - a2 = 400 - 100 = 300

c = √300 = 10√3

Coordinate of foci (0, ± c) = (0, ±10√3)

Statement II is correct.

Statement III:  Length of major axis is 40.

Major axis = 2b = 40

Statement III is correct.

Statement IV: Eccentricity of the ellipse is 

Eccentricity = e = 

e =  = 

Statement IV is correct.

∴ All the statements are correct.

Calculation:

PS + PS' = 2 × 3 √2 

⇒ b² = a² (1 - e²) ⇒ 9 = 18( 1 - e²)

⇒ e = 

Diretrix x = 

⇒ PS.PS' = 

 = 

⇒ (PS. PS')_max = 18 and (PS .PS')_min = 9

Sum = 27

Hence, the correct answer is Option 4.

Concept:

Equation of ellipse: ​

Eccentricity (e) = 

Where, vertices = (± a, 0) and focus = (± ae, 0)

Calculation:

Here, vertices of ellipse (± 5, 0) and foci (±4, 0)

So, a = ±5 ⇒  and

ae = 4 ⇒ e = 4/5

Now, 4/5 = 

∴ Equation of ellipse = 

Hence, option (1) is correct. 

CONCEPT:

The properties of a vertical ellipse  where 0

• Centre of ellipse is (0, 0)
• Vertices of ellipse are: (0, - a) and (0, a)
• Foci of ellipse are: (0, - ae) and (0, ae)
• Length of major axis is 2a
• Length of minor axis is 2b
• Eccentricity of ellipse is given by:  

CALCULATION:

Given: Equation of ellipse is 

As we can see that the given ellipse is a vertical ellipse.

By comparing the given equation of ellipse with  where 0

⇒ a = 6, b = 2

As we know that, eccentricity of ellipse is given by: 

⇒ 

⇒ ae = 4√2

As we know that, foci of a vertical ellipse are: (0, - ae) and (0, ae)

So, the foci of the given ellipse are: (0, - 4√2) and (0, 4√2)

Hence, option D is the correct answer.

Concept:

The distance between the centre and the focus of an ellipse is c = ae

The equation of an ellipse with the length of the major axis 2a and the minor axis 2b is given by: .

Calculation:

Length of the minor axis = 2b = 8.

⇒ b = 4.

Also, c = distance between the centre and the focus = ae = 2.

c2 = a2e2 = a2 - b2

∴ 22 = a2 - 42

⇒ a² = 20

Equation of the ellipse = .

⇒ .

Concept:

Eccentricity: the ratio of the distance between a point on the ellipse and its focus to the distance between the point and the directrix is called its eccentricity.

Calculation:

Let P(x, y) be a point on the ellipse, F(1, 0) be its focus and D = x + y + 1 = 0 be its directrix.

PF = e × PD

⇒ PF2 = e² × PD2

⇒ (x - 1)2 + (y - 0)2 = 

⇒ 4(x² - 2x + 1 + y2) = x2 + y2 + 1 + 2xy + 2x + 2y

⇒ 3x2 + 3y2 - 2xy - 10x - 2y + 3 = 0

Concept Used:

The standard equation of an ellipse with major axis along the x-axis is , where a > b.

The foci of the ellipse are at (±c, 0), where c² = a² - b².

Calculation

Given: The equation of the ellipse is .⇒ a² = 49 and b² = 24

⇒ a = 7 and b =

⇒ c² = a² - b² = 49 - 24 = 25

⇒ c = = 5

⇒ The foci are at (±5, 0).

∴ The foci of the ellipse are (5, 0) and (-5, 0).

The correct option is 4) (5, 0) and (-5, 0).

Calculation

Diameter of this circle

Diameter of this circle

Length of semi-minor axis [along X-axis]

Length of semi major axis [along Y-axis]

Required ellipse

Hence option 3 is correct

Given that and

(here )

equation of ellipse .