Conic Sections — Class 11 Maharashtra HSC Mathematics revision notes
A 5-page visual revision sheet for Conic Sections, written for Class 11 Maharashtra HSC Mathematics. Formulas, diagrams and the traps that cost marks, laid out to be read the night before rather than the term before. Free, and it opens inside CapCompass.
Page 1 as PDF
Conic Sections in one look
one ratio, focus distance over directrix distance, and its value alone decides parabola, ellipse or hyperbola
The Focus-Directrix Definition
- a conic section is the locus of a point P at a fixed distance ratio
- S is a fixed point, the focus; d is a fixed line, the directrix
- that ratio, SP to P's distance from d, stays CONSTANT for every point P
- that constant ratio SP/PM is the eccentricity e; the defining relation is SP=e· PM
- e=1 gives a parabola; 0<e<1 gives an ellipse; e>1 gives a hyperbola
- these curves are literally slices of a double cone
- a plane perpendicular to the axis gives a circle
- one parallel to a generator gives a parabola
- one oblique but not parallel to any generator gives an ellipse
- one parallel to the axis gives a hyperbola
Vocabulary
- axis
- the line about which the conic is symmetric
- vertex
- where the conic meets its own axis
- focal chord
- a chord through the focus
- latus rectum
- the focal chord perpendicular to the axis
- focal distance
- the distance from a point on the conic to the focus
- double ordinate
- a chord through a point on the curve, perpendicular to the axis
Conic Type, by Eccentricity
- e=0 — circle (the special ellipse case)
- 0<e<1 — ellipse
- e=1 — parabola
- e>1 — hyperbola
Standard Parabola: y squared = 4ax
- focus S(a,0), directrix x+a=0, vertex at the origin, with a>0
- setting SP=PM and squaring collapses to y^2=4ax
- the algebra loses every square root because both sides are distances squared
- the curve is symmetric about the X-axis: every x>0 gives two y-values, negatives of each other
- it exists only for x≥0, since y^2≥0 forces 4ax≥0
The Four Standard Parabolas
| Feature | y^2=4ax | x^2=4by |
|---|---|---|
| Focus | (a,0) | (0,b) |
| Directrix | x+a=0 | y+b=0 |
| Axis | y=0 | x=0 |
| Latus rectum ends | (a,±2a) | (±2b,b) |
| Length of latus rectum | 4a | 4b |
| Tangent at vertex | x=0 | y=0 |
| Focal distance of P(x_1,y_1) | x_1+a | y_1+b |
Page 1 of 5. The full sheet, all 5 pages, opens free in the app.
What this Conic Sections sheet covers
- The Focus-Directrix Definition
- Vocabulary
- Conic Type, by Eccentricity
- Standard Parabola: y squared = 4ax
- The Four Standard Parabolas
- Parabola: Parameter, Tangent, Condition of Tangency
- parametric form of y^2=4ax
- tangent at (x_1,y_1)
- tangent at parameter t_1
- condition of tangency, y=mx+c
- Tangents From a Point, and the Directrix
- Standard Ellipse: x squared/a squared + y squared/b squared = 1, a > b
- Ellipse Facts
- Ellipse: Tangent and Condition of Tangency
- tangent at θ_1
- Standard Hyperbola: x squared/a squared - y squared/b squared = 1
- Asymptotes
- Hyperbola: Tangent and Condition of Tangency
- director circle
- Traps that cost marks
- Tricks that save time
How CapCompass uses this sheet
These notes are not a standalone PDF. CapCompass plans Mathematics chapter by chapter against your real exam dates and free hours, and when the plan reaches Conic Sections the sheet opens inside the study block itself — at the page covering the sub-topic that block is on, with the timer still running. Revision arrives at the moment the plan says revise, instead of having to be gone looking for.