Parabola: Essential Concepts for Competitive Exam Success

Parabola

Q. Determine the focus of the parabola defined by the equation x^2 = 12y.

A. (0, 3)
B. (0, -3)
C. (3, 0)
D. (-3, 0)

Solution

The equation x^2 = 4py gives 4p = 12, hence p = 3. The focus is at (0, p) = (0, 3).

Correct Answer: A — (0, 3)

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Q. Determine the focus of the parabola given by the equation x^2 = 8y.

A. (0, 2)
B. (0, 4)
C. (2, 0)
D. (4, 0)

Solution

The standard form of the parabola is x^2 = 4py. Here, 4p = 8, so p = 2. The focus is at (0, p) = (0, 2).

Correct Answer: B — (0, 4)

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Q. Determine the length of the latus rectum of the parabola y^2 = 16x.

A. 4
B. 8
C. 16
D. 32

Solution

The length of the latus rectum for the parabola y^2 = 4px is given by 4p. Here, p = 4, so the length is 16.

Correct Answer: B — 8

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Q. Find the coordinates of the focus of the parabola y^2 = -12x.

A. (-3, 0)
B. (-2, 0)
C. (3, 0)
D. (2, 0)

Solution

The equation y^2 = -12x can be rewritten as (y - 0)^2 = 4p(x - 0) with p = -3, so the focus is at (-3, 0).

Correct Answer: A — (-3, 0)

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Q. Find the directrix of the parabola y^2 = -8x.

A. x = 2
B. x = -2
C. x = 4
D. x = -4

Solution

For the parabola y^2 = 4px, here 4p = -8, so p = -2. The directrix is given by x = -p, which is x = 2.

Correct Answer: B — x = -2

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Q. Find the equation of the parabola that opens downwards with vertex at (0, 0) and passes through the point (2, -4).

A. y = -x^2
B. y = -2x^2
C. y = -1/2x^2
D. y = -4x^2

Solution

Using the vertex form and substituting the point (2, -4), we find that the equation is y = -2x^2.

Correct Answer: B — y = -2x^2

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Q. Find the equation of the parabola with focus at (0, -3) and directrix y = 3.

A. x^2 = -12y
B. x^2 = 12y
C. y^2 = -12x
D. y^2 = 12x

Solution

The distance from the focus to the directrix is 6, so p = -3. The equation is x^2 = 4py, which gives x^2 = -12y.

Correct Answer: A — x^2 = -12y

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Q. Find the equation of the parabola with focus at (0, 2) and directrix y = -2.

A. x^2 = 8y
B. y^2 = 8x
C. y^2 = -8x
D. x^2 = -8y

Solution

The vertex is at (0, 0) and p = 2. The equation is y^2 = 4px, which gives y^2 = 8x.

Correct Answer: A — x^2 = 8y

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Q. Find the equation of the parabola with vertex at (2, 3) and focus at (2, 5).

A. y = (1/4)(x - 2)^2 + 3
B. y = (1/4)(x - 2)^2 - 3
C. y = (1/4)(x + 2)^2 + 3
D. y = (1/4)(x + 2)^2 - 3

Solution

The vertex form of a parabola is given by (x - h)^2 = 4p(y - k). Here, h = 2, k = 3, and p = 1 (distance from vertex to focus). Thus, the equation is (x - 2)^2 = 4(1)(y - 3) or y = (1/4)(x - 2)^2 + 3.

Correct Answer: A — y = (1/4)(x - 2)^2 + 3

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Q. Find the focus of the parabola defined by the equation x^2 = 12y.

A. (0, 3)
B. (0, -3)
C. (3, 0)
D. (-3, 0)

Solution

The equation x^2 = 12y can be rewritten as (y - 0) = (1/3)(x - 0)^2, indicating the focus is at (0, 3).

Correct Answer: A — (0, 3)

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Q. Find the focus of the parabola given by the equation y^2 = 12x.

A. (3, 0)
B. (0, 3)
C. (0, 6)
D. (6, 0)

Solution

The standard form of a parabola is y^2 = 4px. Here, 4p = 12, so p = 3. The focus is at (p, 0) = (3, 0).

Correct Answer: C — (0, 6)

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Q. Find the length of the latus rectum of the parabola y^2 = 16x.

A. 4
B. 8
C. 16
D. 2

Solution

The length of the latus rectum of a parabola y^2 = 4px is given by 4p. Here, 4p = 16, so p = 4. Therefore, the length of the latus rectum is 4 * 4 = 16.

Correct Answer: B — 8

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Q. For the parabola defined by the equation y^2 = 20x, what is the coordinates of the vertex?

A. (0, 0)
B. (5, 0)
C. (0, 5)
D. (10, 0)

Solution

The vertex of the parabola y^2 = 4px is at (0, 0).

Correct Answer: A — (0, 0)

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Q. For the parabola y = x^2 - 4x + 3, find the coordinates of the vertex.

A. (2, -1)
B. (1, 2)
C. (2, 1)
D. (1, -1)

Solution

To find the vertex, use x = -b/(2a). Here, a = 1, b = -4, so x = 2. Substitute x = 2 into the equation to find y = -1. Thus, the vertex is (2, -1).

Correct Answer: A — (2, -1)

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Q. For the parabola y^2 = 16x, what is the coordinates of the vertex?

A. (0, 0)
B. (4, 0)
C. (0, 4)
D. (0, -4)

Solution

The vertex of the parabola y^2 = 4px is at (0, 0).

Correct Answer: A — (0, 0)

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Q. For the parabola y^2 = 20x, what is the coordinates of the vertex?

A. (0, 0)
B. (5, 0)
C. (0, 5)
D. (10, 0)

Solution

The vertex of the parabola y^2 = 4px is at (0, 0). Here, p = 5, but the vertex remains at (0, 0).

Correct Answer: A — (0, 0)

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Q. If the parabola y = ax^2 + bx + c has its vertex at (1, -2), what is the value of a if it passes through the point (0, 0)?

A. 1
B. 2
C. 3
D. 4

Solution

Using the vertex form y = a(x - 1)^2 - 2 and substituting (0, 0), we get 0 = a(0 - 1)^2 - 2 => 2 = a => a = 2.

Correct Answer: B — 2

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Q. If the parabola y^2 = 16x opens to the right, what is the value of p?

A. 2
B. 4
C. 8
D. 16

Solution

In the equation y^2 = 4px, we have 4p = 16, thus p = 4.

Correct Answer: B — 4

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Q. If the parabola y^2 = 20x opens to the right, what is the value of p?

A. 5
B. 10
C. 20
D. 2

Solution

In the equation y^2 = 4px, we have 4p = 20, thus p = 20/4 = 5.

Correct Answer: A — 5

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Q. If the vertex of the parabola y = ax^2 + bx + c is at (1, -2), what is the value of a if b = 4 and c = -6?

A. 1
B. 2
C. 3
D. 4

Solution

The vertex form of a parabola is given by x = -b/(2a). Here, 1 = -4/(2a) => 2a = -4 => a = -2.

Correct Answer: A — 1

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Q. The equation of a parabola is given by x^2 = 16y. What is the length of the latus rectum?

A. 4
B. 8
C. 16
D. 32

Solution

The length of the latus rectum for the parabola x^2 = 4py is given by 4p. Here, 4p = 16, so p = 4. Thus, the length of the latus rectum is 4p = 16.

Correct Answer: B — 8

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Q. The parabola y = -3(x - 2)^2 + 5 opens in which direction?

A. Upwards
B. Downwards
C. Left
D. Right

Solution

Since the coefficient of (x - 2)^2 is negative, the parabola opens downwards.

Correct Answer: B — Downwards

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Q. What is the axis of symmetry for the parabola defined by the equation y^2 = -12x?

A. x = 0
B. y = 0
C. y = -6
D. x = -6

Solution

The axis of symmetry for a parabola in the form y^2 = 4px is the x-axis, which is x = 0.

Correct Answer: A — x = 0

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Q. What is the axis of symmetry for the parabola given by the equation y = -2x^2 + 4x + 1?

A. x = 1
B. y = 1
C. x = 2
D. y = 2

Solution

The axis of symmetry for a parabola in the form y = ax^2 + bx + c is given by x = -b/(2a). Here, a = -2, b = 4, so x = -4/(2*-2) = 1.

Correct Answer: A — x = 1

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Q. What is the axis of symmetry for the parabola given by the equation y = 3x^2 + 6x + 2?

A. x = -1
B. y = -1
C. x = 1
D. y = 1

Solution

The axis of symmetry for a parabola in the form y = ax^2 + bx + c is given by x = -b/(2a). Here, a = 3 and b = 6, so x = -6/(2*3) = -1.

Correct Answer: A — x = -1

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Q. What is the directrix of the parabola given by the equation y^2 = 8x?

A. x = -2
B. x = 2
C. y = -4
D. y = 4

Solution

The standard form of a parabola is (y - k)^2 = 4p(x - h). Here, p = 2, so the directrix is x = -2.

Correct Answer: A — x = -2

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Q. What is the directrix of the parabola y^2 = 8x?

A. x = -2
B. x = 2
C. y = -4
D. y = 4

Solution

For the parabola y^2 = 4px, here 4p = 8, so p = 2. The directrix is x = -p = -2.

Correct Answer: A — x = -2

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Q. What is the equation of the parabola that opens upwards with vertex at the origin and passes through the point (2, 8)?

A. y = 2x^2
B. y = x^2
C. y = 4x^2
D. y = 8x^2

Solution

The vertex form of a parabola is y = ax^2. Since it passes through (2, 8), we have 8 = a(2^2) => 8 = 4a => a = 2. Thus, the equation is y = 4x^2.

Correct Answer: C — y = 4x^2

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Q. What is the equation of the parabola with focus at (0, 2) and directrix y = -2?

A. x^2 = 8y
B. x^2 = -8y
C. y^2 = 8x
D. y^2 = -8x

Solution

The distance from the focus to the directrix is 4, so the equation is y = (1/4)(x - 0)^2 + 0, which simplifies to x^2 = 8y.

Correct Answer: A — x^2 = 8y

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Q. What is the equation of the parabola with focus at (0, 3) and directrix y = -3?

A. x^2 = 12y
B. y^2 = 12x
C. y = 3x^2
D. x = 3y^2

Solution

The distance from the focus to the directrix is 6, so p = 3. The equation is y^2 = 4px = 12y.

Correct Answer: A — x^2 = 12y

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Parabola MCQ & Objective Questions

The concept of a parabola is crucial for students preparing for various school and competitive exams in India. Understanding parabolas not only enhances your mathematical skills but also plays a significant role in scoring well in exams. Practicing MCQs and objective questions on parabolas helps reinforce your understanding and boosts your confidence, making it easier to tackle important questions during your exam preparation.

What You Will Practise Here

Definition and properties of parabolas
Standard form and vertex form of parabolic equations
Graphing parabolas and identifying key features
Applications of parabolas in real-life scenarios
Focus, directrix, and latus rectum of a parabola
Solving quadratic equations related to parabolas
Common transformations of parabolic graphs

Exam Relevance

The topic of parabolas is frequently featured in CBSE, State Boards, NEET, and JEE exams. Students can expect questions that require them to identify the properties of parabolas, solve equations, and graph parabolic functions. Common question patterns include multiple-choice questions that test conceptual understanding and application of formulas related to parabolas.

Common Mistakes Students Make

Confusing the vertex form and standard form of parabolic equations
Misinterpreting the focus and directrix in relation to the parabola
Overlooking the significance of the latus rectum
Making calculation errors while graphing parabolas
Neglecting to check the orientation of the parabola (upward or downward)

FAQs

Question: What is the standard form of a parabola?
Answer: The standard form of a parabola is given by the equation \(y = ax^2 + bx + c\), where 'a' determines the direction and width of the parabola.

Question: How do I find the vertex of a parabola?
Answer: The vertex can be found using the formula \(x = -\frac{b}{2a}\) from the standard form of the equation.

Now is the perfect time to enhance your understanding of parabolas! Dive into our practice MCQs and test your knowledge to excel in your exams. Remember, consistent practice is the key to mastering important Parabola questions for exams!

Parabola

Parabola MCQ & Objective Questions

What You Will Practise Here

Exam Relevance

Common Mistakes Students Make

FAQs

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