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झारखण्ड बोर्ड (JAC) • कक्षा XI • Mathematics • अध्याय 10

अनुमानित समय: 45 Mins

प्रगति: अध्ययनरत

शंकु परिच्छेद (Conic Sections)

In Class 11 Mathematics, "Conic Sections" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.

अध्याय CBT ऑनलाइन टेस्ट दें सभी अध्याय देखें

प्रस्तावना अध्याय अवधारणाएँ अध्याय सारांश स्व-मूल्यांकन अभ्यास टेस्ट दें

🪐 Have You Ever Wondered?

Why do all planets orbit the Sun in ellipses rather than perfect circles, and why are satellite dishes curved into parabolas to focus faint television...

Why do all planets orbit the Sun in ellipses rather than perfect circles, and why are satellite dishes curved into parabolas to focus faint television signals from deep space? Conic sections are slices of a double cone that dictate gravitational celestial mechanics.

यह अध्याय क्यों महत्वपूर्ण है

In Class 11 Mathematics, "Conic Sections" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.

अध्ययन से पूर्व (आवश्यक ज्ञान)

Circle geometry from Class 9 & 10.
Cartesian plane.
Distance formula.

इस अध्याय के लक्ष्य

Define and derive standard equation of a Circle: $(x - h)^2 + (y - k)^2 = r^2$.
Define Parabola with focus, directrix, and latus rectum; analyze standard forms ($y^2 = 4ax, x^2 = 4ay$).
Define Ellipse with major/minor axes, foci, eccentricity ($e < 1$), and latus rectum: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.
Define Hyperbola with transverse/conjugate axes, foci, eccentricity ($e > 1$): $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$.
Relate focal distance and eccentricity: $c = ae$ and $b^2 = a^2(1 - e^2)$ for ellipses.

अध्याय रूपरेखा एवं प्रगति

1 1. Circle & Parabola

2 2. Ellipse ($e < 1$)

3 3. Hyperbola ($e > 1$)

सम्पूर्ण सैद्धांतिक एवं वैचारिक अध्ययन

1. Circle & Parabola

Circle: Set of all points equidistant from center $(h, k)$: $$\mathbf{(x - h)^2 + (y - k)^2 = r^2}$$
Parabola: Set of points equidistant from focus $(a, 0)$ and directrix $x = -a$: $$\mathbf{y^2 = 4ax} \quad (\text{Length of Latus Rectum} = 4a)$$

2. Ellipse ($e < 1$)

Set of points whose sum of distances from two fixed foci is constant ($2a$): $$\mathbf{\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1} \quad (a > b) \quad \text{with} \quad \mathbf{c^2 = a^2 - b^2}$$ Eccentricity: $\mathbf{e = \frac{c}{a} < 1}$. Foci at $(\pm c, 0)$; Latus Rectum $= \frac{2b^2}{a}$.

3. Hyperbola ($e > 1$)

Set of points whose difference of distances from two fixed foci is constant ($2a$): $$\mathbf{\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1} \quad \text{with} \quad \mathbf{c^2 = a^2 + b^2}$$ Eccentricity: $\mathbf{e = \frac{c}{a} > 1}$. Foci at $(\pm c, 0)$; Latus Rectum $= \frac{2b^2}{a}$.

चित्रात्मक व्याख्या एवं मॉडल

Conic Sections Master Matrix

Conceptual framework, core mechanisms, and analytical relationships

Academic Architecture

1. Circle & Parabola • 2. Ellipse ($e < 1$)

अध्याय का सार संक्षेप एवं 10 मुख्य निष्कर्ष

मुख्य बिंदु 1

Eccentricity Metric: $e = 0$ Circle, $e = 1$ Parabola, $0 < e < 1$ Ellipse, $e > 1$ Hyperbola.

मुख्य बिंदु 2

Parabolic Reflection: Parallel incoming rays reflect through the single focus point (satellite dishes).

मुख्य बिंदु 3

Kepler's First Law: Planetary orbits are ellipses with the Sun at one focus.

मुख्य बिंदु 4

Latus Rectum: Chord through a focus perpendicular to the principal symmetry axis.

मुख्य बिंदु 5

Focal Property: Difference of focal distances remains invariant ($2a$) on hyperbolas.

स्व-मूल्यांकन अभ्यास (Check Your Understanding)

मूल वैचारिक स्पष्टता की जांच के लिए नैदानिक प्रश्न। पहले स्वयं हल करें, फिर उत्तर देखें।

Find the center and radius of the circle: $x^2 + y^2 - 4x - 8y - 45 = 0$.

उत्तर एवं व्याख्या देखें

उत्तर: Complete squares: $(x - 2)^2 - 4 + (y - 4)^2 - 16 - 45 = 0 \implies (x - 2)^2 + (y - 4)^2 = 65$. Center is $(2, 4)$; Radius is $\sqrt{65}$.
Center (2, 4), radius √65.

Find the coordinates of the focus, equation of the directrix, and length of latus rectum for $y^2 = 12x$.

उत्तर एवं व्याख्या देखें

उत्तर: Compare with $y^2 = 4ax \implies 4a = 12 \implies a = 3$. Focus: $(3, 0)$. Directrix: $x = -3$. Length of latus rectum: $4a = 12$.
Focus (3, 0), directrix x = -3, LR = 12.

Find the coordinates of the foci, vertices, and eccentricity of the ellipse: $\frac{x^2}{25} + \frac{y^2}{9} = 1$.

उत्तर एवं व्याख्या देखें

उत्तर: $a^2 = 25, b^2 = 9 \implies a = 5, b = 3$. $c = \sqrt{25 - 9} = 4$. Foci: $(\pm 4, 0)$. Vertices: $(\pm 5, 0)$. Eccentricity $e = c/a = 4/5 = 0.8$.
Foci (±4, 0), vertices (±5, 0), e = 0.8.

Find the equation of the hyperbola with foci $(\pm 5, 0)$ and transverse axis of length 8.

उत्तर एवं व्याख्या देखें

उत्तर: Transverse axis $2a = 8 \implies a = 4$. Foci $(\pm c, 0) \implies c = 5$. $b^2 = c^2 - a^2 = 25 - 16 = 9$. Equation: $\frac{x^2}{16} - \frac{y^2}{9} = 1$.
x^2 / 16 - y^2 / 9 = 1.

What is the geometric definition of a parabola?

उत्तर एवं व्याख्या देखें

उत्तर: The locus of a point that moves such that its distance from a fixed point (focus) is equal to its perpendicular distance from a fixed straight line (directrix).
Equidistant from focus and directrix.

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