परवलय (Parabola) की उत्केंद्रता (Eccentricity) कितनी होती है? | Agrigyan

Q.1

ICAR AIEEA PG 2024 - Statistical Sciences (Q.94)

MCQ (Single Correct) +4 −1

In hyperbola, the length of the transverse axis is the distance between:

अतिपरवलय (hyperbola) में, अनुप्रस्थ अक्ष (transverse axis) की लंबाई किसके बीच की दूरी होती है?

two vertices दोनों शीर्षों के बीच की दूरी two foci दोनों नाभियों के बीच की दूरी vertex and the origin शीर्ष और मूल बिंदु के बीच की दूरी focus and the vertex नाभि और शीर्ष के बीच की दूरी

Q.2

ICAR AIEEA PG 2024 - Statistical Sciences (Q.95)

MCQ (Single Correct) +4 −1

The center of the circle $4x^2 + 4y^2 - 8x + 12y - 25 = 0$ is:

वृत्त $4x^2 + 4y^2 - 8x + 12y - 25 = 0$ का केंद्र है:

(-2, 3) (-2, 3) (1, -3/2) (1, -3/2) (-4, 6) (-4, 6) (4, -6) (4, -6)

Q.3

ICAR AIEEA PG 2024 - Statistical Sciences (Q.96)

MCQ (Single Correct) +4 −1

The length of the latus rectum of $x^2 = -9y$ is equal to:

परवलय $x^2 = -9y$ के नाभिलंब (latus rectum) की लंबाई किसके बराबर है?

3 3 -3 -3 9/4 9/4 9 9

Q.4

ICAR AIEEA PG Statistical Sciences 09 July 2023 (Shift 1) Paper (Q.90)

MCQ (Single Correct) +4 −1

The eccentricity of the parabola is :

परवलय (Parabola) की उत्केंद्रता (Eccentricity) कितनी होती है?

1 1 0 0 Less than 1 1 से कम Greater than 1 1 से अधिक

Q.5

ICAR AIEEA PG Statistical Sciences 09 July 2023 (Shift 1) Paper (Q.91)

MCQ (Single Correct) +4 −1

Let $x^2 = \frac{25}{2}y$ be the equation of a parabola, then the equation of its directrix is given by :

माना कि $x^2 = \frac{25}{2}y$ एक परवलय (parabola) का समीकरण है, तो इसकी नियता (directrix) का समीकरण होगा :

$y = \frac{25}{2}$ $y = \frac{25}{2}$ $y = -\frac{25}{4}$ $y = -\frac{25}{4}$ $y = -\frac{25}{8}$ $y = -\frac{25}{8}$ $y = \frac{25}{8}$ $y = \frac{25}{8}$

The standard upward-opening parabola is $x^2 = 4ay$.
Comparing $x^2 = \frac{25}{2}y$ with $x^2 = 4ay$ gives:
$4a = \frac{25}{2} \implies a = \frac{25}{8}$.
The directrix of $x^2 = 4ay$ is horizontal and located below the vertex at $y = -a$.

Therefore, the equation of the directrix is $y = -\frac{25}{8}$. Detailed analysis of options:

$y = \frac{25}{2}$ represents the coefficient of $y$, not the directrix.
$y = -\frac{25}{4}$ corresponds to taking $a = 25/4$, which forgets the factor of 4.
$y = -\frac{25}{8}$ is correct ($y = -a = -25/8$).
$y = \frac{25}{8}$ is the equation of the line passing through the focus ($y = a$), not the directrix.

ऊपर की ओर खुलने वाले मानक परवलय (parabola) का समीकरण $x^2 = 4ay$ होता है।
दिए गए समीकरण $x^2 = \frac{25}{2}y$ की तुलना $x^2 = 4ay$ से करने पर:
$4a = \frac{25}{2} \implies a = \frac{25}{8}$.
परवलय $x^2 = 4ay$ की नियता (directrix) का समीकरण $y = -a$ होता है।

अतः नियता का समीकरण $y = -\frac{25}{8}$ है। विकल्पों का विस्तृत विश्लेषण:

$y = \frac{25}{2}$ गलत है, यह $y$ का गुणांक है।
$y = -\frac{25}{4}$ गलत है।
$y = -\frac{25}{8}$ सही उत्तर है ($y = -a = -25/8$)।
$y = \frac{25}{8}$ नाभि (focus) से गुजरने वाली रेखा का समीकरण है, नियता का नहीं।

Explanation not matching this question? क्या व्याख्या इस प्रश्न से मेल नहीं खा रही? Report

सक्रिय अध्याय: Coordinate Geometry: Straight Lines, Circles, Parabola, Ellipse and Hyperbola in Two Dimensions

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