Analytical geometry on NET is about straight lines and triangles on the coordinate plane. About a dozen formulas cover it: distance, slope, section, the forms of a line, distance from a point to a line, angle between lines, and translation and rotation of axes. You never need to draw to scale.
Key facts
- Mathematics is 50% of NET-Engineering and NET-Natural Sciences, and 30% of NET-Architecture
- Analytical Geometry is unit 5 of the Class 12 PECTAA mathematics book (National Curriculum 2023)
- Our chapter map estimates about 7 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 puts lines and triangles on Cartesian coordinates and answers geometry questions with algebra. It uses distance, slope, division of a segment, and translation and rotation of axes. Circles, parabolas and the other curves come in the next chapter, conic sections.
NET questions here are short. A typical one gives two points or a line and asks for a slope, an equation, a distance or a ratio. The work is plugging into the right formula without a sign error.
Formulas
Points and segments:
distance d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
midpoint ((x1 + x2)/2, (y1 + y2)/2)
section m:n ((m x2 + n x1)/(m + n), (m y2 + n y1)/(m + n)) internal
centroid ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3)
area (1/2) |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|
area = 0 means the three points are collinear
Lines:
slope m = (y2 - y1)/(x2 - x1) = tan(inclination)
slope of ax + by + c = 0 m = -a/b
point-slope y - y1 = m(x - x1)
two-intercept x/a + y/b = 1
normal form x cos α + y sin α = p
parallel m1 = m2
perpendicular m1 m2 = -1
angle from l1 to l2 tan θ = (m2 - m1)/(1 + m1 m2)
Distances involving lines:
point to line |a x0 + b y0 + c| / sqrt(a^2 + b^2)
parallel lines |c1 - c2| / sqrt(a^2 + b^2)
(same a and b in both equations)
Change of axes:
translation, new origin (h, k): X = x - h, Y = y - k
rotation by θ: X = x cos θ + y sin θ
Y = -x sin θ + y cos θ
Worked MCQs
Five questions modelled on our practice bank. We solved each one and checked it by putting the answer back into the question.
1. Find the line through (-3, 5) perpendicular to x + 2y + 3 = 0.
The given line has slope -1/2, so the perpendicular slope is 2:
y - 5 = 2(x + 3)
2x - y + 11 = 0
2. Find the distance of (-7, 13) from 2x + y + 13 = 0.
|2(-7) + 13 + 13| / sqrt(4 + 1) = |12| / sqrt(5) = 12/sqrt(5)
3. The origin is shifted to (-4, -6). Where does P(-6, -8) move to?
X = -6 - (-4) = -2
Y = -8 - (-6) = -2
Answer: (-2, -2). The usual mistake is to add the shift instead of subtracting it.
4. In what ratio does the y-axis divide the segment from (-3, -4) to (1, -2)?
Every point on the y-axis has x = 0. Put that into the section formula with ratio m:n:
(m(1) + n(-3)) / (m + n) = 0
m = 3n, so the ratio is 3:1
A faster way: the ratio is the distances of the two x-coordinates from zero, 3 and 1.
5. Find the distance between y = 2x + 4 and 3y = 6x - 5.
Both lines have slope 2, so they are parallel. Write them with the same a and b:
2x - y + 4 = 0
2x - y - 5/3 = 0
distance = |4 + 5/3| / sqrt(5) = (17/3)/sqrt(5) = 17/(3 sqrt(5))
Skip the rewrite and you will divide by sqrt(45) for one line and sqrt(5) for the other. That gives one of the wrong options.
Common traps
- Slope from the general form. For
ax + by + c = 0the slope is-a/b, with the minus sign. Dropping it flips perpendicular and parallel answers. - Inclination of a negative slope. Slope -1 means an inclination of 135°, not -45°. Inclination is measured from 0° to 180°.
- Parallel-line distance with unequal coefficients. Scale one equation first, as in example 5.
- Internal vs external division. If the ratio comes out negative, the division is external. Do not drop the sign.
- Collinear checks. Equal slopes between two pairs of points prove collinearity. For (5, -2), (1, 2) and (-2, 5) both slopes are -1, so the three points lie on one line.
How to prepare
Learn the formula blocks above until you can write them from memory, then do a timed set of 20. This chapter rewards accuracy more than insight, so most lost marks are sign errors. Write the formula out in full on your rough sheet before substituting.
The maths formula sheet has these on one page, and the NET mathematics guide shows where this chapter fits in a full plan.
FAQ
Is analytical geometry hard in NET?
No. It is formula-driven and most questions need one formula and one substitution. Speed and sign accuracy decide your score here.
What is the difference between analytical geometry and conic sections?
Analytical geometry covers points, straight lines and triangles. Conic sections covers circles, parabolas, ellipses and hyperbolas. They are separate units in the Class 12 book.
How do I find the angle between two lines quickly?
Find both slopes and use tan θ = (m2 - m1)/(1 + m1 m2). If 1 + m1 m2 = 0 the lines are perpendicular and the angle is 90°.
