Parabola Basics for Competitive Exam Success

Parabola Basics

Q. For the parabola defined by the equation x^2 = -12y, what is the direction in which it opens?

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

Solution

The equation x^2 = -12y indicates that the parabola opens downwards.

Correct Answer: C — Left

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Q. For the parabola defined by the equation x^2 = 16y, what is the distance from the vertex to the focus?

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

Solution

In the equation x^2 = 4py, we have 4p = 16, thus p = 4. The distance from the vertex to the focus is 4.

Correct Answer: B — 4

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Q. For the parabola defined by the equation x^2 = 16y, what is the length of the latus rectum?

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

Solution

The length of the latus rectum for the parabola x^2 = 4py is 4p. Here, p = 4, so the length is 8.

Correct Answer: B — 8

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Q. For the parabola defined by the equation y = -x^2 + 4x - 3, what is the y-intercept?

A. -3
B. 0
C. 3
D. 4

Solution

To find the y-intercept, set x = 0. The equation becomes y = -0^2 + 4(0) - 3 = -3.

Correct Answer: A — -3

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Q. If a parabola has its vertex at (3, -2) and opens downwards, what is the general form of its equation?

A. y + 2 = a(x - 3)^2
B. y + 2 = -a(x - 3)^2
C. y - 2 = a(x + 3)^2
D. y - 2 = -a(x + 3)^2

Solution

For a downward-opening parabola with vertex (h, k), the equation is y - k = -a(x - h)^2. Here, h = 3 and k = -2.

Correct Answer: B — y + 2 = -a(x - 3)^2

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Q. If a parabola opens to the left, which of the following is its standard form?

A. y^2 = -4px
B. x^2 = -4py
C. y^2 = 4px
D. x^2 = 4py

Solution

The standard form of a parabola that opens to the left is y^2 = -4px.

Correct Answer: A — y^2 = -4px

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Q. If a parabola opens to the right and has its vertex at the origin, what is the general form of its equation?

A. y^2 = 4px
B. x^2 = 4py
C. y = mx + c
D. x = ay^2

Solution

The general form of a parabola that opens to the right is y^2 = 4px.

Correct Answer: A — y^2 = 4px

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

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

Solution

The distance from the focus to the directrix is 4, so the equation is x^2 = 8y.

Correct Answer: A — x^2 = 8y

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Q. The equation of a parabola with vertex at (0, 0) and directrix y = -3 is?

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

Solution

The distance from the vertex to the directrix is 3, so the equation is x^2 = -12y.

Correct Answer: A — x^2 = -12y

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Q. The equation of a parabola with vertex at (0, 0) and focus at (0, 3) is?

A. x^2 = 12y
B. y^2 = 12x
C. x^2 = 6y
D. y^2 = 6x

Solution

The distance from the vertex to the focus is 3, so the equation is x^2 = 12y.

Correct Answer: A — x^2 = 12y

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Q. The parabola y^2 = 12x opens in which direction?

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

Solution

The equation y^2 = 12x indicates that the parabola opens to the right since it is in the form y^2 = 4px.

Correct Answer: D — Right

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Q. The vertex of the parabola given by the equation y = 2x^2 - 4x + 1 is located at which point?

A. (1, -1)
B. (2, 0)
C. (1, 0)
D. (0, 1)

Solution

To find the vertex, use the formula x = -b/(2a). Here, a = 2, b = -4, so x = 1. Plugging x = 1 into the equation gives y = -1.

Correct Answer: A — (1, -1)

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Q. What is the axis of symmetry for the parabola given by the equation y = -3(x - 2)^2 + 5?

A. x = 2
B. y = 5
C. y = -3
D. x = -2

Solution

The axis of symmetry for a parabola in vertex form y = a(x - h)^2 + k is x = h. Here, h = 2.

Correct Answer: A — x = 2

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

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

Solution

The axis of symmetry for the parabola y^2 = 4px is the x-axis.

Correct Answer: B — y-axis

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Q. What is the directrix of the parabola defined by the equation y^2 = 20x?

A. x = -5
B. x = 5
C. y = 5
D. y = -5

Solution

For the equation y^2 = 4px, p = 5. The directrix is given by x = -p, which is x = -5.

Correct Answer: A — x = -5

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Q. What is the equation of the directrix of the parabola x^2 = 12y?

A. y = 3
B. y = -3
C. y = 6
D. y = -6

Solution

The directrix of the parabola x^2 = 4py is given by y = -p. Here, p = 3, so the directrix is y = -3.

Correct Answer: B — y = -3

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Q. What is the focus of the parabola defined by the equation y^2 = 20x?

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

Solution

In the equation y^2 = 4px, we have 4p = 20, thus p = 5. The focus is at (5, 0).

Correct Answer: A — (5, 0)

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Q. What is the focus of the parabola given by the equation y^2 = 20x?

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

Solution

For the parabola y^2 = 4px, here 4p = 20, so p = 5. The focus is at (5, 0).

Correct Answer: A — (5, 0)

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Q. What is the latus rectum of the parabola given by the equation y^2 = 12x?

A. 3
B. 4
C. 6
D. 12

Solution

The latus rectum of a parabola y^2 = 4px is given by 4p. Here, 4p = 12, so p = 3, and the latus rectum is 4p = 12.

Correct Answer: C — 6

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Q. What is the value of p for the parabola defined by the equation y^2 = 20x?

A. 5
B. 10
C. 20
D. 4

Solution

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

Correct Answer: A — 5

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

A. (1, -1)
B. (1, 0)
C. (2, 1)
D. (0, 1)

Solution

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

Correct Answer: A — (1, -1)

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Q. Which of the following points lies on the parabola y = x^2 - 4?

A. (2, 0)
B. (0, -4)
C. (1, -3)
D. (3, 5)

Solution

Substituting x = 1 into the equation gives y = 1^2 - 4 = -3, so the point (1, -3) lies on the parabola.

Correct Answer: C — (1, -3)

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Q. Which of the following points lies on the parabola y^2 = 8x?

A. (2, 4)
B. (1, 2)
C. (4, 4)
D. (2, 2)

Solution

To check if a point lies on the parabola, substitute the x-coordinate into the equation. For (2, 4), 4^2 = 16 and 8*2 = 16, so it lies on the parabola.

Correct Answer: A — (2, 4)

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Parabola Basics MCQ & Objective Questions

Understanding the basics of parabolas is crucial for students preparing for school exams and competitive tests. Mastering Parabola Basics not only enhances your conceptual clarity but also boosts your confidence in tackling MCQs and objective questions. Regular practice with these important questions can significantly improve your exam scores and overall performance.

What You Will Practise Here

Definition and properties of parabolas
Standard form and vertex form of a parabola
Graphical representation and sketching of parabolas
Focus, directrix, and axis of symmetry
Applications of parabolas in real-life scenarios
Common equations and formulas related to parabolas
Solving objective questions and MCQs on parabolas

Exam Relevance

Parabola Basics is a significant topic in various examinations, including CBSE, State Boards, NEET, and JEE. Questions related to parabolas often appear in the form of multiple-choice questions, where students are required to identify properties, sketch graphs, or solve equations. Familiarity with common question patterns will help you tackle these effectively and efficiently.

Common Mistakes Students Make

Confusing the standard form and vertex form of a parabola
Misinterpreting the focus and directrix concepts
Errors in graphing parabolas accurately
Overlooking the significance of the axis of symmetry
Failing to apply the correct formulas in problem-solving

FAQs

Question: What is the standard form of a parabola?
Answer: The standard form of a parabola is given by the equation y = ax² + 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 = -b/(2a) from the standard form of the parabola.

Ready to enhance your understanding of Parabola Basics? Dive into our practice MCQs and test your knowledge today! Consistent practice will pave the way for success in your exams.

Parabola Basics

Parabola Basics MCQ & Objective Questions

What You Will Practise Here

Exam Relevance

Common Mistakes Students Make

FAQs

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