Mathematics
Conic Sections
285 Questions
Conic sections deal with the geometry of curves like ellipses, hyperbolas, and parabolas formed by the intersection of a plane with a cone. This is an important topic in advanced mathematics sections of competitive exams. Practice these questions to understand eccentricity, focal distances, and equations of tangents and normals.
Ellipse equationsHyperbola eccentricityFocal distance calculationsTangent and normal equationsConic standard forms
Conic Sections Questions
Which conic section is represented by the equation (x^2 + y^2 = r^2)?
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Circle
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Ellipse
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Parabola
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Hyperbola
A
Correct answer
Explanation
The equation (x^2 + y^2 = r^2) represents a circle with radius (r) and center at the origin.
What is the eccentricity of a circle?
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0
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1
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\(\sqrt{2}\)
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\(\sqrt{3}\)
A
Correct answer
Explanation
The eccentricity of a circle is 0 because it is a perfectly round shape with no elongation.
Which conic section is represented by the equation (\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1)?
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Circle
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Ellipse
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Parabola
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Hyperbola
B
Correct answer
Explanation
The equation (\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1) represents an ellipse with semi-major axis (a) and semi-minor axis (b).
What is the eccentricity of an ellipse?
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0
-
1
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\(\sqrt{2}\)
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Between 0 and 1
D
Correct answer
Explanation
The eccentricity of an ellipse is a value between 0 and 1 that determines how elongated the ellipse is.
What is the eccentricity of a parabola?
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0
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1
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\(\sqrt{2}\)
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Undefined
D
Correct answer
Explanation
The eccentricity of a parabola is undefined because it is an open curve that extends infinitely.
Which conic section is represented by the equation (\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1)?
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Circle
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Ellipse
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Parabola
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Hyperbola
D
Correct answer
Explanation
The equation (\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1) represents a hyperbola with transverse axis along the (x)-axis and conjugate axis along the (y)-axis.
What is the eccentricity of a hyperbola?
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0
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1
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Greater than 1
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Between 0 and 1
C
Correct answer
Explanation
The eccentricity of a hyperbola is a value greater than 1 that determines how elongated the hyperbola is.
What is the standard form of the equation of an ellipse?
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\(x^2 + y^2 = r^2\)
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\(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)
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\(y^2 = 4px\)
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\(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)
B
Correct answer
Explanation
The standard form of the equation of an ellipse is (\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1), where (a) and (b) are the lengths of the semi-major and semi-minor axes, respectively.
What is the equation of the directrix of an ellipse with center at the origin and semi-major axis (a)?
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\(x = -a\)
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\(x = a\)
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\(y = -a\)
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\(y = a\)
B
Correct answer
Explanation
The equation of the directrix of an ellipse with center at the origin and semi-major axis (a) is (x = a).
What is the eccentricity of a binary star system?
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The degree to which the orbit of the two stars is elliptical
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The degree to which the orbit of the two stars is inclined to the plane of the sky
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The degree to which the orbit of the two stars is precessing
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The degree to which the orbit of the two stars is nutating
A
Correct answer
Explanation
The eccentricity of a binary star system is the degree to which the orbit of the two stars is elliptical.
What is the general formula for the Madhava series for the circumference of an ellipse?
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$$C = 4aE(e) = 4a \left(1 - \frac{e^2}{2} + \frac{e^4}{2\cdot4} - \frac{e^6}{2\cdot4\cdot6} + \cdots\right)$$
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$$C = 4aE(e) = 4a \left(1 - \frac{e^2}{2} + \frac{e^4}{2\cdot4} - \frac{e^6}{2\cdot4\cdot6} + \cdots\right)$$
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$$C = 4aE(e) = 4a \left(1 - \frac{e^2}{2} + \frac{e^4}{2\cdot4} - \frac{e^6}{2\cdot4\cdot6} + \cdots\right)$$
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$$C = 4aE(e) = 4a \left(1 - \frac{e^2}{2} + \frac{e^4}{2\cdot4} - \frac{e^6}{2\cdot4\cdot6} + \cdots\right)$$
A,B,C,D
Correct answer
Explanation
The general formula for the Madhava series for the circumference of an ellipse is $$C = 4aE(e) = 4a \left(1 - \frac{e^2}{2} + \frac{e^4}{2\cdot4} - \frac{e^6}{2\cdot4\cdot6} + \cdots\right)$$.
What is the formula for calculating the obliquity of the ecliptic?
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Obliquity = (23.4392911 - (0.0000004 * t))
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Obliquity = (23.4392911 + (0.0000004 * t))
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Obliquity = (23.4392911 - (0.0000004 * t^2))
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Obliquity = (23.4392911 + (0.0000004 * t^2))
A
Correct answer
Explanation
The formula for calculating the obliquity of the ecliptic is Obliquity = (23.4392911 - (0.0000004 * t)), where t is the number of years since the beginning of the Kali Yuga.
What is the formula for calculating the nutation in obliquity?
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Nutation in Obliquity = (9.2105 * cos(Ω) + 0.5532 * sin(Ω) - 0.0904 * cos(2Ω))
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Nutation in Obliquity = (9.2105 * sin(Ω) + 0.5532 * cos(Ω) - 0.0904 * sin(2Ω))
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Nutation in Obliquity = (9.2105 * cos(Ω) - 0.5532 * sin(Ω) - 0.0904 * cos(2Ω))
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Nutation in Obliquity = (9.2105 * sin(Ω) - 0.5532 * cos(Ω) - 0.0904 * sin(2Ω))
A
Correct answer
Explanation
The formula for calculating the nutation in obliquity is Nutation in Obliquity = (9.2105 * cos(Ω) + 0.5532 * sin(Ω) - 0.0904 * cos(2Ω)), where Ω is the longitude of the ascending node of the Moon.
What is the equation of the ellipse?
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$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$
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$$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$
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$$\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$$
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$$\frac{x^2}{b^2} - \frac{y^2}{a^2} = 1$$
A
Correct answer
Explanation
The equation of the ellipse is $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$, where $a$ is the semi-major axis and $b$ is the semi-minor axis.
What is the eccentricity of an ellipse?
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The ratio of the semi-major axis to the semi-minor axis
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The ratio of the semi-minor axis to the semi-major axis
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The square of the ratio of the semi-major axis to the semi-minor axis
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The square of the ratio of the semi-minor axis to the semi-major axis
D
Correct answer
Explanation
The eccentricity of an ellipse is the square of the ratio of the semi-minor axis to the semi-major axis.