Conic sections on NET test the standard equations of the circle, parabola, ellipse and hyperbola, the link between a, b, c and the eccentricity, and the tangent conditions. Put each curve in standard form and most of these MCQs answer themselves.
Key facts
- Mathematics is 50% of NET-Engineering and NET-Natural Sciences, and 30% of NET-Architecture
- Conic Section is unit 6 of the Class 12 PECTAA mathematics book (National Curriculum 2023)
- Our chapter map estimates about 8 MCQs from this chapter. NUST does not publish chapter counts
Source: NUST subject weightings, verified 22 August 2026; SailToNUST chapter map
What the chapter covers
The chapter defines each conic as a locus, a set of points obeying a distance rule. A circle keeps a fixed distance from its centre. A parabola, ellipse and hyperbola keep a fixed ratio e between the distance to a focus and the distance to a directrix. From those definitions come the standard equations, then tangents and normals.
Almost every NET question here starts with one move: complete the square to get the curve into standard form. After that you read off the centre, a, b, c, the foci or the eccentricity.
Formulas
Circle:
(x - h)^2 + (y - k)^2 = r^2
x^2 + y^2 + 2gx + 2fy + c = 0
centre (-g, -f), r = sqrt(g^2 + f^2 - c)
r^2 = 0 point circle, r^2 negative imaginary circle
tangent to x^2 + y^2 = a^2 at (x1, y1): x x1 + y y1 = a^2
y = mx + c touches x^2 + y^2 = a^2 if c^2 = a^2 (1 + m^2)
Parabola y^2 = 4ax:
focus (a, 0), directrix x = -a, latus rectum 4a, e = 1
tangent at (x1, y1): y y1 = 2a(x + x1)
y = mx + c is a tangent if c = a/m
Ellipse x^2/a^2 + y^2/b^2 = 1 with a greater than b:
c^2 = a^2 - b^2, e = c/a (less than 1)
foci (±c, 0), vertices (±a, 0)
sum of distances to the foci = 2a
latus rectum = 2b^2/a
y = mx + c is a tangent if c^2 = a^2 m^2 + b^2
Hyperbola x^2/a^2 - y^2/b^2 = 1:
c^2 = a^2 + b^2, e = c/a (greater than 1)
foci (±c, 0)
difference of distances to the foci = 2a
asymptotes y = ±(b/a) x
y = mx + c is a tangent if c^2 = a^2 m^2 - b^2
Eccentricity sorts the family: e = 0 for a circle, e less than 1 for an ellipse, e = 1 for a parabola, e greater than 1 for a hyperbola.
Worked MCQs
Five questions of the kind our practice bank is full of. We solved and rechecked each one.
1. Find the eccentricity of x^2/4 - y^2/9 = 1.
It is a hyperbola, so add: c^2 = 4 + 9 = 13. Then e = c/a = sqrt(13)/2.
2. Find the centre of 2x^2 + 2y^2 + 2x - 4y + 7 = 0.
Divide by 2 first. The circle form needs the x^2 coefficient to be 1:
x^2 + y^2 + x - 2y + 7/2 = 0
g = 1/2, f = -1 centre (-1/2, 1)
r^2 = 1/4 + 1 - 7/2 = -9/4
The centre is (-1/2, 1). Note the radius squared is negative, so this is an imaginary circle. If a question asks about its radius, that is the point.
3. If y = 4x + c touches x^2/8 + y^2/4 = 1, find c.
c^2 = a^2 m^2 + b^2 = 8(16) + 4 = 132
c = ±sqrt(132)
4. Find the tangent to y^2 = 6x at the point where y = 6.
On the curve, y = 6 gives x = 6. Here 4a = 6, so a = 3/2:
y(6) = 2(3/2)(x + 6)
6y = 3x + 18
x - 2y + 6 = 0
5. Find the distance between the foci of 9x^2 - 18x + 4y^2 + 8y - 23 = 0.
Complete the square in x and y:
9(x - 1)^2 - 9 + 4(y + 1)^2 - 4 - 23 = 0
9(x - 1)^2 + 4(y + 1)^2 = 36
(x - 1)^2/4 + (y + 1)^2/9 = 1
The larger denominator is under y, so the major axis is vertical, with a^2 = 9 and b^2 = 4. Then c^2 = 5, and the foci are 2c = 2 sqrt(5) apart.
Common traps
- Ellipse vs hyperbola signs. Ellipse:
c^2 = a^2 - b^2. Hyperbola:c^2 = a^2 + b^2. They are the easiest pair to mix up in this chapter. - Not dividing through. A circle equation with
2x^2 + 2y^2must be divided by 2 before you read g, f and c. - Which axis is major. For an ellipse, a sits under the larger denominator, whether that is x or y.
- Is it a circle at all? The coefficients of
x^2andy^2must be equal and there must be noxyterm.(4a - 3)x^2 + a y^2 + ...is a circle only when4a - 3 = a, soa = 1. - Point position. Substitute the point into
x^2 + y^2 - r^2. Positive means outside, zero means on, negative means inside. (5, 5) andx^2 + y^2 = 25gives 25, so the point is outside.
How to prepare
Learn the four formula blocks as one table: same layout, different signs. Then practise completing the square until it is automatic, because nearly every question starts there. Straight lines from analytical geometry come up inside conic questions too, especially for tangents and chords.
The formula sheet puts the four conics side by side if you want a single page to revise from.
FAQ
Which conic sections topics are most important for NET?
Standard forms, eccentricity, foci and tangent conditions. The circle has the most questions in our practice bank, so start there.
How do I identify a conic from its general equation?
With no xy term: equal x^2 and y^2 coefficients give a circle, only one squared term gives a parabola, same signs give an ellipse, and opposite signs give a hyperbola.
What is the eccentricity of a circle?
Zero. An ellipse has e between 0 and 1, a parabola has e = 1, and a hyperbola has e greater than 1.
