What went wrong when the king rode under the arch?

It was too low and knocked off his crown

How does the chain of blame unfold in the poem?

Each accused person blames another, ending with the king's own changes

Find the centre and radius of \(x^2+y^2+8x+10y-8=0\).

Completing squares: \((x+4)^2+(y+5)^2=49\); centre \((-4,-5)\), radius \(7\).

A parabola is the set of all points in a plane equidistant from a fixed line (directrix) and a fixed point (focus) not on the line.

For \(y^2=8x\), give the focus, directrix and latus rectum.

Comparing with \(y^2=4ax\), \(a=2\); focus \((2,0)\), directrix \(x=-2\), latus rectum \(4a=8\).

Define an ellipse and state the relation among \(a,b,c\).

An ellipse is the set of points the sum of whose distances from two foci is constant; for \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\), \(c^2=a^2-b^2\).

Give the eccentricity and latus rectum of an ellipse.

Eccentricity \(e=\dfrac{c}{a}<1\); latus rectum \(=\dfrac{2b^2}{a}\).

For \(\dfrac{x^2}{25}+\dfrac{y^2}{9}=1\), find the foci and eccentricity.

\(a=5,\ b=3,\ c=\sqrt{25-9}=4\); foci \((\pm4,0)\), eccentricity \(\dfrac45\).

Define a hyperbola and state the relation among \(a,b,c\).

A hyperbola is the set of points the difference of whose distances from two foci is constant; for \(\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1\), \(c^2=a^2+b^2\).

For \(\dfrac{x^2}{9}-\dfrac{y^2}{16}=1\), give the foci and eccentricity.

\(a=3,\ b=4,\ c=\sqrt{9+16}=5\); foci \((\pm5,0)\), eccentricity \(\dfrac53\).

What is the latus rectum of an ellipse and a hyperbola?

For both, the length of the latus rectum is \(\dfrac{2b^2}{a}\), measured perpendicular to the major/transverse axis through a focus.

CBSE · Senior secondary

CBSE Class 11 — Notes, Chapters & Practice Quizzes

Q: For \(\dfrac{x^2}{25}+\dfrac{y^2}{9}=1\), find the foci and eccentricity.

\(a=5,\ b=3,\ c=\sqrt{25-9}=4\); foci \((\pm4,0)\), eccentricity \(\dfrac45\).

Q: Define a hyperbola and state the relation among \(a,b,c\).

A hyperbola is the set of points the difference of whose distances from two foci is constant; for \(\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1\), \(c^2=a^2+b^2\).

Q: For \(\dfrac{x^2}{9}-\dfrac{y^2}{16}=1\), give the foci and eccentricity.

\(a=3,\ b=4,\ c=\sqrt{9+16}=5\); foci \((\pm5,0)\), eccentricity \(\dfrac53\).

Q: What is the latus rectum of an ellipse and a hyperbola?

For both, the length of the latus rectum is \(\dfrac{2b^2}{a}\), measured perpendicular to the major/transverse axis through a focus.

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Chapter 10: Conic Sections — Class 11 Mathematics

Mathematics · 14 chapters

Summary, key terms, important questions and a practice quiz with AI diagnosis for each.

Chapter 10: Conic Sections

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The equation of a circle with centre \((h,k)\) and radius \(r\) is:

Summary

Conic sections are the curves obtained when a plane cuts a double-napped right circular cone: a circle, ellipse, parabola or hyperbola according to the angle of the cut, with degenerate cases (a point, a line, a pair of lines) when the plane passes through the vertex. A circle is the set of points at fixed distance (radius) from a centre, with equation \((x-h)^2+(y-k)^2=r^2\). A parabola is the set of points equidistant from a focus and a directrix; the standard form \(y^2=4ax\) has focus \((a,0)\), directrix \(x=-a\) and latus rectum \(4a\). An ellipse is the set of points whose distances from two foci have constant sum; \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\) with \(c^2=a^2-b^2\), eccentricity \(e=c/a<1\) and latus rectum \(\dfrac{2b^2}{a}\). A hyperbola is the set of points whose distances from two foci have constant difference; \(\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1\) with \(c^2=a^2+b^2\), eccentricity \(e=c/a>1\) and latus rectum \(\dfrac{2b^2}{a}\). These curves model planetary orbits, reflectors, arches and suspension cables.

Sections of a coneCircle and its equationParabola, focus, directrix and latus rectumEllipse, foci, eccentricity and latus rectumHyperbola, foci, eccentricity and latus rectumStandard equations and applications

Key terms

Conic section

A curve formed by intersecting a plane with a double-napped right circular cone.

Circle

Set of points at a fixed distance \(r\) from a centre; \((x-h)^2+(y-k)^2=r^2\).

Parabola

Set of points equidistant from a focus and a directrix; standard \(y^2=4ax\).

Ellipse

Set of points whose distances from two foci have constant sum; \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\).

Hyperbola

Set of points whose distances from two foci have constant difference; \(\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1\).

Eccentricity

\(e=c/a\): \(e=0\) circle, \(01\) hyperbola.

Important questions

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Conic section

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A curve formed by intersecting a plane with a double-napped right circular cone.

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Practice quiz · Conic Sections

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Conic Sections

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