JEE Maths — Coordinate Geometry Complete Chapter Guide

Chapter Overview & Weightage

Coordinate Geometry is one of the most reliable scoring chapters in JEE Main. Students who get this chapter right consistently pull 12-16 marks without breaking a sweat. The chapter spans Straight Lines, Circles, and Conics — and each section has its own personality in terms of how questions are framed.

JEE Main consistently drops 4-5 questions from Coordinate Geometry — that’s roughly 16-20 marks per attempt. In 2024, both Session 1 and Session 2 had 5 questions each. This chapter alone can swing your percentile significantly.

Year-by-Year Weightage (JEE Main)

Year	Straight Lines	Circles	Parabola	Ellipse	Hyperbola	Total Qs
2024 (S1)	1	1	1	1	1	5
2024 (S2)	2	1	1	1	0	5
2023 (S1)	1	2	1	0	1	5
2023 (S2)	1	1	2	1	0	5
2022 (S1)	2	1	1	1	0	5
2022 (S2)	1	1	1	1	1	5
2021	1	2	1	1	0	5

The pattern is clear: Circles and Straight Lines together account for 2-3 questions every year without fail. Parabola shows up almost every session. Ellipse appears ~70% of the time. Hyperbola is bonus — when it comes, it’s usually the easier question of the set.

Key Concepts You Must Know

Prioritised by how often they actually appear in PYQs — not by NCERT chapter order.

Circles (High Priority)

Equation of circle: standard form and general form — converting between them
Circle through three points
Equation of tangent at a point and from an external point
Condition for two circles to be orthogonal: $2g_1g_2 + 2f_1f_2 = c_1 + c_2$
Family of circles through intersection of two circles: $S_1 + \lambda S_2 = 0$
Common chord of two circles (subtract the two equations directly)

Straight Lines (High Priority)

Distance of a point from a line — used everywhere, including in conics
Angle bisectors of two lines — pay attention to the $\pm$ sign selection
Family of lines through intersection: $L_1 + \lambda L_2 = 0$
Locus problems — this is where JEE gets creative

Parabola (Medium-High)

Standard forms: $y^2 = 4ax$ , $y^2 = -4ax$ , $x^2 = 4ay$ , $x^2 = -4ay$
Parametric form $(at^2, 2at)$ — most PYQ solutions run through this
Chord of contact, chord with given midpoint
Normal at a point and condition for normal to pass through a given point

Ellipse (Medium)

Relationship $b^2 = a^2(1 - e^2)$ — the foundation of every ellipse problem
Auxiliary circle and eccentric angle
Director circle: $x^2 + y^2 = a^2 + b^2$
Sum of focal distances = $2a$ (constant property, appears in MCQs)

Hyperbola (Lower Priority, but don’t ignore)

Asymptotes: $\frac{x}{a} \pm \frac{y}{b} = 0$
Rectangular hyperbola $xy = c^2$ with parametric form $(ct, c/t)$
Difference of focal distances = $2a$

Important Formulas

d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}

When to use: Anytime you need to find the radius of an inscribed circle, verify tangency, or work with angle bisectors. This formula appears in at least 1 question every JEE session — often hidden inside a circle or locus problem.

Length of tangent $= \sqrt{S_1}$ where $S_1 = x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c$

When to use: Whenever the problem says “tangent drawn from point P” — don’t try to find the actual tangent line unless asked. Just compute $\sqrt{S_1}$ .

For $y^2 = 4ax$ : point is $(at^2, 2at)$

Chord joining $t_1$ and $t_2$ : $y(t_1 + t_2) = 2x + 2at_1t_2$

Normal at $t$ : $y + tx = 2at + at^3$

When to use: Whenever the question mentions “chord”, “normal”, or “tangent” on a parabola. Parametric approach cuts the algebra by half.

Line $y = mx + c$ is tangent to:

$y^2 = 4ax$ if $c = \frac{a}{m}$
$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ if $c^2 = a^2m^2 + b^2$
$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ if $c^2 = a^2m^2 - b^2$

When to use: “Find the value of k if the line … is tangent to …” — this is the direct formula, no quadratic discrimination needed (though both methods work).

From external point $(x_1, y_1)$ , chord of contact to $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is:

\frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1

Same pattern for circle: $T = 0$ (replace $x^2 \to xx_1$ , $y^2 \to yy_1$ , $x \to \frac{x+x_1}{2}$ , etc.)

When to use: “Two tangents are drawn from P… find the chord of contact.” Direct substitution, no solving required.

Solved Previous Year Questions

PYQ 1 — JEE Main 2024 Session 1

Q: The locus of the foot of perpendicular drawn from the centre of the ellipse $x^2 + 3y^2 = 6$ on any tangent to it is $(x^2 + y^2)^2 = ax^2 + by^2$ . Find $a + b$ .

Solution:

The ellipse is $\frac{x^2}{6} + \frac{y^2}{2} = 1$ , so $a^2 = 6$ , $b^2 = 2$ .

A tangent to this ellipse in slope form: $y = mx \pm \sqrt{6m^2 + 2}$

Let the foot of perpendicular from origin $(0,0)$ to this tangent be $(h, k)$ .

The perpendicular from origin to a tangent $y = mx + c$ has slope $-\frac{1}{m}$ , so:

\frac{k}{h} \cdot m = -1 \implies m = -\frac{h}{k}

Also, $(h, k)$ lies on the tangent: $k = mh + c$ , so $c = k - mh = k + \frac{h^2}{k} = \frac{h^2 + k^2}{k}$

Now use the tangency condition $c^2 = 6m^2 + 2$ :

\frac{(h^2+k^2)^2}{k^2} = 6 \cdot \frac{h^2}{k^2} + 2

(h^2 + k^2)^2 = 6h^2 + 2k^2

So $a = 6$ , $b = 2$ , giving $a + b = \mathbf{8}$ .

The foot-of-perpendicular locus technique is a standard JEE pattern. The key step is always: express $m$ in terms of $(h,k)$ using the perpendicularity condition, then substitute into the tangency condition.

PYQ 2 — JEE Main 2023 Session 2

Q: If the line $y = mx + c$ is a common tangent to the hyperbola $\frac{x^2}{100} - \frac{y^2}{64} = 1$ and the circle $x^2 + y^2 = 36$ , find the value of $c^2$ .

Solution:

For the line to be tangent to the hyperbola $\frac{x^2}{100} - \frac{y^2}{64} = 1$ :

c^2 = 100m^2 - 64 \quad \cdots (1)

For the line to be tangent to the circle $x^2 + y^2 = 36$ (radius = 6):

\frac{|c|}{\sqrt{1+m^2}} = 6 \implies c^2 = 36(1 + m^2) \quad \cdots (2)

From (1) and (2):

100m^2 - 64 = 36 + 36m^2

64m^2 = 100 \implies m^2 = \frac{25}{16}

Substituting back in (2):

c^2 = 36\left(1 + \frac{25}{16}\right) = 36 \cdot \frac{41}{16} = \frac{369}{4}

$c^2 = \mathbf{\dfrac{369}{4}}$

A very common error here: students apply $c^2 = a^2m^2 + b^2$ for hyperbola — but the tangency condition for hyperbola is $c^2 = a^2m^2 - b^2$ (minus, not plus). Mixing this up with ellipse costs marks every year.

PYQ 3 — JEE Main 2022 Session 1

Q: Let the normal at point P on the parabola $y^2 = 16x$ meet the x-axis at Q. If M is the midpoint of PQ, find the locus of M.

Solution:

Take $P = (at^2, 2at)$ with $a = 4$ , so $P = (4t^2, 8t)$ .

Normal at $P$ on $y^2 = 4ax$ meets x-axis at $Q = (2a + at^2, 0)$ .

With $a = 4$ : $Q = (8 + 4t^2, 0)$ .

Midpoint $M = (h, k)$ :

h = \frac{4t^2 + 8 + 4t^2}{2} = 4t^2 + 4

k = \frac{8t + 0}{2} = 4t

From the second equation: $t = \frac{k}{4}$

Substituting in the first:

h = 4 \cdot \frac{k^2}{16} + 4 = \frac{k^2}{4} + 4

k^2 = 4(h - 4) \implies y^2 = 4(x-4)

This is a parabola with vertex $(4, 0)$ — the locus is $\mathbf{y^2 = 4(x-4)}$ .

Difficulty Distribution

For JEE Main (based on 2021-2024 PYQs):

Subtopic	Easy	Medium	Hard
Straight Lines	40%	50%	10%
Circles	30%	55%	15%
Parabola	25%	55%	20%
Ellipse	20%	60%	20%
Hyperbola	35%	50%	15%

The “Hard” questions in Coordinate Geometry are usually locus problems or questions involving two conics simultaneously. If you’re targeting 90+ percentile, those are the ones to master. For 80+ percentile, nailing the Easy and Medium questions is enough — that’s 3-4 questions guaranteed.

Expert Strategy

Week 1 — Foundations: Lock Straight Lines and Circles completely. These two subtopics have the highest return on investment — conceptually simpler, but JEE questions require knowing when to use which formula.

Week 2 — Conics: Do Parabola with heavy focus on parametric methods. Parabola has the most formula-to-concept connections; once you nail parametric form, 70% of parabola PYQs become routine.

Week 3 — Ellipse & Hyperbola + Integration: Cover Ellipse focusing on eccentricity, focal distances, and director circle. Hyperbola is lighter — asymptotes and rectangular hyperbola are the hot topics.

The smartest approach toppers use: solve every Coordinate Geometry question from JEE Main 2019-2024 (all sessions). That’s roughly 50-60 questions. You’ll see that most “new” questions are just the same core idea dressed differently. Pattern recognition is the actual skill being tested.

Time allocation during the exam: Coordinate Geometry questions are usually solvable in 2-3 minutes if you recognise the pattern. If a question is taking more than 4 minutes, mark it and return — coordinate geometry almost never requires brute-force algebra if you’re using the right formula.

Don’t skip locus problems. Many students avoid locus because it feels open-ended. But locus problems follow a strict 4-step algorithm: assume the moving point as $(h, k)$ , write the given condition, eliminate the parameter, replace $(h, k)$ with $(x, y)$ . That’s it.

Common Traps

Trap 1 — Forgetting the $\pm$ in tangent length. When the question asks for the equation of tangent from an external point, there are always two tangents. Many students write only one and lose half the marks (or the MCQ option doesn’t match).

Trap 2 — Applying ellipse formula to hyperbola. The tangency condition for $y = mx + c$ : ellipse uses $c^2 = a^2m^2 + b^2$ , hyperbola uses $c^2 = a^2m^2 - b^2$ . These get swapped under exam pressure constantly.

Trap 3 — Not converting to standard form first. If the circle/conic is not in standard form, convert before using any formula. Applying $S_1 = 0$ directly on a non-standard equation gives wrong results. JEE setters deliberately give equations in general form for this reason.

Trap 4 — Sign of $a$ in parabola. $y^2 = 4ax$ opens right (a > 0). $y^2 = -4ax$ opens left. When the parabola is given as $y^2 = -12x$ , the focus is at $(-3, 0)$ , not $(3, 0)$ . This mistake appeared in JEE Main 2023 and many students got the direction wrong.

Trap 5 — Family of circles $S_1 + \lambda S_2 = 0$ always passes through intersection points. But $S_1 + \lambda S_2 = 0$ where $\lambda = -1$ gives the common chord, not a circle. Students sometimes substitute $\lambda = -1$ thinking they’re getting a circle from the family — they’re not.

One final note on JEE Advanced: If you’re targeting Advanced, add these to your list — pole and polar, pair of tangents ( $SS_1 = T^2$ ), and the chord of contact derivation from first principles. JEE Advanced tests the why behind formulas, not just their application.