Class 11 Maths · Original EZT practice
Conic Sections (Class 11 Chapter 10): summary and practice
Conic Sections cuts a double cone with a plane: a circle when the plane is level, a parabola at the edge angle, an ellipse between, a hyperbola past it. Each curve gets a standard equation and a defining length. All questions are original EZT tutor practice following the chapter's own section order.
6 original worked examples, 6 fresh self-check questions and 4 mixed questions, with explained answers. These are not copied textbook exercises, an official chapter index or a full answer key. No affiliation with NCERT is claimed.
Official sources for topic and edition context, checked 21 October 2026:
1. Sections of a cone
Slice a double right-circular cone and the cut edge is a conic: circle, ellipse, parabola or hyperbola, decided by the plane's angle against the cone's slant.
2. The circle
Every point at a fixed distance r from a centre (h, k): (x - h)^2 + (y - k)^2 = r^2. The general form x^2 + y^2 + 2gx + 2fy + c = 0 has centre (-g, -f) and radius sqrt(g^2 + f^2 - c).
3. The parabola
Points equally far from a focus and a directrix. Standard: y^2 = 4ax, focus (a, 0), directrix x = -a, latus rectum 4a.
4. The ellipse
Sum of distances to two foci is constant: x^2/a^2 + y^2/b^2 = 1 with a > b. Eccentricity e = c/a < 1 where c^2 = a^2 - b^2; latus rectum 2b^2/a.
5. The hyperbola
Difference of distances to two foci is constant: x^2/a^2 - y^2/b^2 = 1. Eccentricity e = c/a > 1 where c^2 = a^2 + b^2; latus rectum 2b^2/a.
Choose, solve, check, then try a fresh question.
Before calculating, read the method prompt and name the condition that makes it useful. Write your own attempt. Open the steps only when you need them, and use the check to catch a wrong sign, unit or interpretation. Finish with the fresh question before reading its answer.
Read a circle's centre and radius
Choose the method: Complete the square on x and y.
EZT practice question
Centre and radius of x^2 + y^2 - 4x + 6y - 12 = 0?
Show the worked solution
- Group: (x^2 - 4x) + (y^2 + 6y) = 12.
- Complete: (x - 2)^2 - 4 + (y + 3)^2 - 9 = 12.
- (x - 2)^2 + (y + 3)^2 = 25.
Answer: Centre (2, -3), radius 5.
Check: g = -2, f = 3, c = -12: centre (-g, -f) = (2, -3) and r = sqrt(4 + 9 + 12) = 5 - matches.
Watch for this: The centre is (-g, -f), not (g, f) - the signs flip.
Now try it yourself
Radius of x^2 + y^2 + 8x - 10y - 8 = 0?
Check your answer
7.
sqrt(16 + 25 + 8) = sqrt(49).
Work with y^2 = 4ax
Choose the method: Read a off the coefficient; focus (a, 0), directrix x = -a.
EZT practice question
Focus, directrix and latus rectum of y^2 = 12x?
Show the worked solution
- 4a = 12, so a = 3.
- Focus (3, 0), directrix x = -3.
- Latus rectum 4a = 12.
Answer: Focus (3, 0), directrix x = -3, latus rectum 12.
Check: A point on the curve, say (3, 6), is 6 from the focus and 6 from the directrix - equal, as the definition demands.
Watch for this: a is a QUARTER of the coefficient, not the coefficient itself.
Now try it yourself
Focus of y^2 = 8x?
Check your answer
(2, 0).
4a = 8 gives a = 2.
Ellipse axes and eccentricity
Choose the method: Bigger denominator is a^2; c^2 = a^2 - b^2; e = c/a.
EZT practice question
For x^2/25 + y^2/9 = 1 find the foci, eccentricity and latus rectum.
Show the worked solution
- a^2 = 25, b^2 = 9, so c^2 = 16 and c = 4.
- Foci (4, 0) and (-4, 0); e = 4/5.
- Latus rectum 2b^2/a = 18/5.
Answer: Foci (4, 0) and (-4, 0), e = 0.8, latus rectum 3.6.
Check: e < 1 as every ellipse must be, and 2b^2/a = 18/5 = 3.6 - consistent.
Watch for this: For an ellipse c^2 is a^2 MINUS b^2 - the hyperbola is the one that adds.
Now try it yourself
Eccentricity of x^2/16 + y^2/7 = 1?
Check your answer
3/4.
c^2 = 9, c = 3, a = 4.
Hyperbola axes and eccentricity
Choose the method: Positive term gives a^2; c^2 = a^2 + b^2; e = c/a.
EZT practice question
For x^2/16 - y^2/9 = 1 find the foci, eccentricity and latus rectum.
Show the worked solution
- a^2 = 16, b^2 = 9, so c^2 = 25 and c = 5.
- Foci (5, 0) and (-5, 0); e = 5/4.
- Latus rectum 2b^2/a = 18/4 = 9/2.
Answer: Foci (5, 0) and (-5, 0), e = 1.25, latus rectum 4.5.
Check: e > 1 as every hyperbola must be - 5/4 qualifies.
Watch for this: Here c^2 is a^2 PLUS b^2 - using minus shrinks the foci.
Now try it yourself
Eccentricity of x^2/9 - y^2/16 = 1?
Check your answer
5/3.
c^2 = 25, c = 5, a = 3.
Write a circle's equation
Choose the method: Plug centre (h, k) and r into (x - h)^2 + (y - k)^2 = r^2.
EZT practice question
Circle with centre (-1, 2) and radius 3?
Show the worked solution
- h = -1, k = 2, r = 3.
- (x + 1)^2 + (y - 2)^2 = 9.
- Expand if required: x^2 + y^2 + 2x - 4y - 4 = 0.
Answer: (x + 1)^2 + (y - 2)^2 = 9.
Check: The centre (-1, 2) is exactly 3 from, say, (2, 2) on the circle: sqrt(9) = 3.
Watch for this: The bracket holds (x - h): a centre of -1 makes it (x + 1).
Now try it yourself
Circle with centre (0, 0) and radius 7?
Check your answer
x^2 + y^2 = 49.
h = k = 0, r^2 = 49.
Identify the conic from its equation
Choose the method: One square and a product: parabola. Same-sign equal squares: circle. Same-sign unequal: ellipse. Opposite signs: hyperbola.
EZT practice question
Name the conics: (a) x^2 = 8y (b) x^2/4 + y^2/9 = 1 (c) x^2 - y^2 = 1.
Show the worked solution
- (a) One squared variable: parabola.
- (b) Same signs, unequal denominators: ellipse.
- (c) Opposite signs: hyperbola.
Answer: (a) parabola, (b) ellipse, (c) hyperbola.
Check: Each matches its standard form: y^2 = 4ax in disguise, x^2/a^2 + y^2/b^2 = 1, and x^2/a^2 - y^2/b^2 = 1.
Watch for this: x^2 + y^2 = 9 is a circle, not an ellipse - the ellipse needs UNEQUAL denominators.
Now try it yourself
Identify x^2/16 + y^2/16 = 1.
Check your answer
A circle of radius 4.
Equal coefficients: x^2 + y^2 = 16.
Try a mixed set
Choose a method or explanation before opening each answer.
1. Circle equation, centre (h, k)?
Show answer and reason
(x - h)^2 + (y - k)^2 = r^2.
Every point at distance r.
2. Parabola standard?
Show answer and reason
y^2 = 4ax, focus (a, 0), latus rectum 4a.
Equal distance to focus and directrix.
3. Ellipse c^2?
Show answer and reason
a^2 - b^2 (a > b).
e = c/a < 1.
4. Hyperbola c^2?
Show answer and reason
a^2 + b^2.
e = c/a > 1.
Try it yourself
Work these on paper before checking any notes. Answers are deliberately not shown here.
- Centre and radius of x^2 + y^2 - 4x + 6y - 12 = 0.
- Focus of y^2 = 12x.
- Eccentricity of x^2/25 + y^2/9 = 1.
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Questions about this practice page
- Why are these curves called conic sections?
- All four appear as the cut edge when a plane slices a double cone. A level cut gives a circle, a tilted cut an ellipse, a cut parallel to the slant a parabola, and a steeper cut meeting both halves of the cone a hyperbola.
- How do I tell ellipse and hyperbola formulas apart?
- Sign of the second term: plus means ellipse with c^2 = a^2 - b^2, minus means hyperbola with c^2 = a^2 + b^2. The ellipse closes into an oval with e < 1; the hyperbola opens into two branches with e > 1.
- What is the latus rectum?
- The chord through the focus parallel to the directrix (for a parabola) or perpendicular to the major axis (ellipse, hyperbola). Its length is 4a for y^2 = 4ax and 2b^2/a for the other two.
- How do I find a circle's centre from the general equation?
- For x^2 + y^2 + 2gx + 2fy + c = 0, the centre is (-g, -f) and the radius is sqrt(g^2 + f^2 - c). Equivalently, complete the square on x and y separately.
- Are these the NCERT exercise questions for Conic Sections (Class 11)?
- No. EZT pages are original practice following the chapter's section order - use them to prepare for the textbook exercises, not as a substitute for your book.