10. Conic Sections

11 topics2 h

Chapter 10 of NCERT Mathematics for Class 11: Conic Sections. Three sections — Slicing a double cone, Circle and parabola and Ellipse and hyperbola. 11 videos, 2 h in all.

Worked answers to the 70 exercise questions in this chapter: Exercise 10.1 · Exercise 10.2 · Exercise 10.3 · Exercise 10.4 · Miscellaneous Exercise

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Part 1

Slicing a double cone

12 minOne surface, four curves, chosen by the angle of the cutCircle, ellipse, parabola and hyperbola look like four separate topics. They are one surface, a double cone, read at four settings of a single number: the angle of the cutting plane.12 min · 13 parts

12 minCuts through the vertex, where the curve degeneratesSlide a cone's cutting plane down until it passes through the vertex, and the four familiar curves collapse to three: a point, one line, or two lines crossing. Nothing has gone wrong.12 min · 13 parts

Part 2

Circle and parabola

13 minFixed distance from a fixed point, turned into an equationA parabola's definition names a point and a line; an ellipse's and a hyperbola's each name two points. A circle's names one point and one number - nothing left to derive.13 min · 13 partsहिंदी

16 minBalancing a point against a line, and the four equations that resultA circle needs one point and one number, and never cares where the axes sit. A parabola needs a point and a line instead, and how that pair sits against the axes changes everything.16 min · 14 parts

12 minThe chord through the focus that measures how open the curve isA parabola's defining length, called a, cannot be pointed to on the curve - it runs from the vertex to a focus sitting off the curve entirely. The latus rectum can actually be measured, and equals 4a.12 min · 12 parts

Part 3

Ellipse and hyperbola

12 minTwo fixed points and a fixed total distanceAn ellipse asks two distances from a moving point to add to one fixed total - but only if that total exceeds the distance between the two fixed points themselves.12 min · 12 parts

14 minThe right triangle hidden in the figure, and the single number that sets the shapeThe identity linking an ellipse's three lengths looks like Pythagoras borrowed straight from a right triangle, and it is not. It comes from computing the ellipse's own defining sum at two points and forcing agreement.14 min · 12 parts

16 minPutting the centre at the origin, and reading the axes off the equationSlide a point along an ellipse and its distance to each focus changes at a steady rate, one growing exactly as fast as the other shrinks - why the total never moves.16 min · 15 parts

12 minReplacing the fixed total by a fixed differenceChange one word in the ellipse's definition - sum becomes difference - and the sentence means nothing until a direction is chosen: which focus counts as farther.12 min · 13 parts

16 minOnce c outgrows a, the ellipse's eccentricity and equation become the hyperbola'sThe hyperbola's equation is not a fresh derivation. It is the ellipse's, run again with one comparison between two lengths reversed - enough to turn a loop into two branches.16 min · 15 parts

14 minThe latus rectum measured on an open curve, by the ellipse's own calculationAn ellipse's latus rectum and a hyperbola's reduce to the identical formula - no coincidence, since both curves' eccentricities cancel out of the working entirely.14 min · 14 parts

What you will be able to do

Describe how the double-napped right circular cone is generated, and name the role of each of the two lines involved
State the condition §10.2.1 imposes on the cutting plane, and say why §10.2.2 is the complementary case rather than an exception
State Definition 1 and name its two ingredients
State Definition 2 and name its two ingredients and the exclusion it carries
State Definition 3 and identify the three conditions a latus rectum must satisfy
State Definition 4 and name its two ingredients and the quantity held constant
Compute the sum of focal distances at a vertex of the ellipse and show it equals the major axis length
State the two frame decisions §10.5.3 makes and say what each one is worth
State Definition 7 and identify what is held constant
State Definition 8 and compute e for a given hyperbola
State Definition 6 and Definition 9 and say what the two have in common

What this chapter assumes you already know

What it means for a line to be inclined to another line at a fixed angle, and that rotating one line about another sweeps a surface
That a plane and a surface can meet in a curve, and that the curve depends on how the plane is placed
Degree measure, and comparison of angles by size — the whole classification is three inequalities and two equalities
That a plane and a line either meet at one point, or the plane contains the whole line, or they never meet — the trichotomy the whole section runs on
Reading a strict inequality against an inclusive one, and noticing when an endpoint changes hands
The distance formula for two points in the plane, which §9.1 of Chapter 9 (p. 151) recalls and this section uses without re-deriving
Expanding a squared binomial, and recognising an expansion when it is only partly present
Solving a small simultaneous system in three unknowns

It builds directly on Steepness as the tangent of an angle, and the one line that has none.

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

"A circle is one of the ellipses."
"β is how much the plane is tilted from horizontal."
"The cone is a solid."
"The cone stops at the base circle."
"The parabola is a range of tilts."
"Two branches means the plane was used twice."
"Any plane through the cone gives one of these four."
"Degenerate means the maths went wrong."