Ellipse Practice Questions - NDA Mathematics Question Bank

Question 1 of 5 NDA MCQ

What is the distance between the foci of the ellipse x² + 2y² = 1?

Detailed Explanation

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Explanation:

Given, equation ellipse is

x² + 2y² = 1

⇒ x²/1 + y²/(1/√2)² = 1

∴ a = 1 and b = 1/√2

We know that

b² = a² − c²

⇒ 1/2 = 1 − c²

⇒ c² = 1/2

⇒ c = ±1/√2

Distance between the foci = |2c|

= 2 × 1/√2 = √2

Question 2 of 5 NDA MCQ

Let P(x, y) be any point on the ellipse 25x² + 16y² = 400. If Q(0, 3) and R(0, −3) are two points, then what is (PQ + PR) equal to?

Detailed Explanation

Solution

Option (b) is correct.

Explanation:

The given ellipse is represented by the equation:

\[ \frac{x^2}{16} + \frac{y^2}{25} = 1 \]

The points \( Q(0, 3) \) and \( R(0, -3) \) are vertically aligned with the center of the ellipse at the origin \( (0,0) \). The lengths of the semi-major axis \( a = 5 \) and semi-minor axis \( b = 4 \) can be derived from the coefficients in the ellipse equation. Using the distance formula, the distances \( PQ \) and \( PR \) from a point \( P(x,y) \) on the ellipse to points \( Q \) and \( R \) are given by: \[ PQ = \sqrt{(x - 0)^2 + (y - 3)^2} \] \[ PR = \sqrt{(x - 0)^2 + (y + 3)^2} \] To find \( PQ + PR \): \[ PQ + PR = \sqrt{x^2 + (y - 3)^2} + \sqrt{x^2 + (y + 3)^2} \] To minimize this expression, we use the property of ellipses: the sum of the distances from any point on the ellipse to the foci remains constant and equal to \( 2a \). Here, the foci of the ellipse can be computed as: \[ c = \sqrt{a^2 - b^2} = \sqrt{25 - 16} = 3 \] So, the foci are at \( (c, 0) \) and \( (-c, 0) \), specifically \( (3,0) \) and \( (-3,0) \). The total distance, \( PQ + PR \), becomes \( 2 \times 5 = 10 \). Therefore, the final result is:

\[ PQ + PR = 10 \]

📌 Hints / Properties Used:

Distance formula: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
Sum of distances from points on an ellipse to its foci is constant.
Ellipse properties: \( c = \sqrt{a^2 - b^2} \)

📊 Visual Diagram Suggestion: The diagram should illustrate the ellipse centered at the origin, highlighting the foci at points \( (3,0) \) and \( (-3,0) \). Show point P on the ellipse and draw distances \( PQ \) and \( PR \) to points Q and R (0, 3) and (0, -3), respectively.

📖 Factual Verification & Reference:

Verified against NCERT Class XI Mathematics, Chapter 11 (Conic Sections).

Question 3 of 5 NDA MCQ

For the next two (02) items:(From 57Q to 58Q)

P(x, y) is any point on the ellipse

x² + 4y² = 1.

Let E and F be the foci of the ellipse.

What is PE + PF equal to?

Detailed Explanation

Question 4 of 5 NDA MCQ

Consider the following points :

Which of the above points lie on latus rectumof ellipse -

Which of the above points lie on latus rectum

of ellipse ?

Detailed Explanation

Question 5 of 5 NDA MCQ

Consider the following in respect of the equation

1. The equation represents an ellipse if k = 19.
2. The equation represents a hyperbola if k = 12.
3. The equation represents a circle if k = 20.

How many of the statements given above are correct?

Detailed Explanation

Ellipse

Solution

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