Coordinate Geometry

Concept Core

Lines, circles, and conics — the complete coordinate geometry framework.

Straight Lines — All Forms

Form	Equation	Use When
Slope-intercept	y = mx + c	Know slope m and y-intercept c
Point-slope	y − y₁ = m(x − x₁)	Know slope m and one point
Two-point	(y−y₁)/(y₂−y₁) = (x−x₁)/(x₂−x₁)	Know two points
Intercept form	x/a + y/b = 1	Know both intercepts
Normal form	x cosα + y sinα = p	Know perp distance p and angle α

Distance from (x₁,y₁) to ax+by+c=0: d = |ax₁+by₁+c|/√(a²+b²) Angle between lines: tan θ = |(m₁−m₂)/(1+m₁m₂)|

Circles

Standard form: (x−h)² + (y−k)² = r² (centre (h,k), radius r)

General form: x² + y² + 2gx + 2fy + c = 0

Centre = (−g, −f), radius = √(g²+f²−c)

Condition: for real circle, g²+f²−c > 0

Tangent from external point: length = √(x₁²+y₁²+2gx₁+2fy₁+c)

Parabola

Standard forms:

y² = 4ax: opens right, focus (a,0), directrix x = −a y² = −4ax: opens left x² = 4ay: opens up, focus (0,a) x² = −4ay: opens down

Vertex at origin. Axis of symmetry along x or y. Latus rectum = 4a (chord through focus ⊥ axis).

Ellipse

Standard form (a > b): x²/a² + y²/b² = 1

b² = a²(1−e²) e = eccentricity (0 < e < 1 for ellipse) Foci: (±ae, 0) Directrices: x = ±a/e Length of latus rectum = 2b²/a

Hyperbola

Standard form: x²/a² − y²/b² = 1

b² = a²(e²−1) e > 1 for hyperbola Foci: (±ae, 0) Asymptotes: y = ±(b/a)x

Rectangular hyperbola: xy = c² (asymptotes are the axes)

Key Relationships — Conics Unified

All conics are sections of a cone. General equation: ax²+2hxy+by²+2gx+2fy+c = 0

Circle: a=b, h=0. Parabola: h²=ab. Ellipse: h²<ab. Hyperbola: h²>ab.

Eccentricity: circle e=0, ellipse 0<e<1, parabola e=1, hyperbola e>1.

Formula Vault

Every coordinate geometry formula for EAPCET.

Distance Formula

d = √((x₂−x₁)²+(y₂−y₁)²)

Between two points

Section Formula

P = (mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)

Divides in ratio m:n internally

Slope

m = (y₂−y₁)/(x₂−x₁) = tanθ

θ = angle with positive x-axis

Perp. Lines

m₁m₂ = −1

Product of slopes = −1

Point to Line Distance

|ax₁+by₁+c|/√(a²+b²)

Line: ax + by + c = 0

Circle General Form

x²+y²+2gx+2fy+c = 0

Centre (−g,−f); r=√(g²+f²−c)

Tangent Length to Circle

√(x₁²+y₁²+2gx₁+2fy₁+c)

From external point (x₁,y₁)

Parabola y²=4ax

Focus (a,0); Directrix x=−a

Latus rectum = 4a

Ellipse Eccentricity

e = √(1−b²/a²)

0 < e < 1; b² = a²(1−e²)

Hyperbola Asymptotes

y = ±(b/a)x

For x²/a² − y²/b² = 1

Area of Triangle

½|x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂)|

From three vertices

Pair of Lines

ax²+2hxy+by²=0

tanθ=2√(h²−ab)/(a+b)

Worked Examples

5 problems — line distance, circle, parabola, ellipse, and a classic trap.

EasyDistance from (3,4) to line 3x−4y+5=0▾

Find the perpendicular distance from point (3,4) to line 3x − 4y + 5 = 0.

d = |ax₁+by₁+c|/√(a²+b²) = |3(3)−4(4)+5|/√(9+16)

= |9−16+5|/√25 = |−2|/5 = 2/5

✓ Distance = 2/5

EasyFind centre and radius of x²+y²−4x+6y−3=0▾

Find the centre and radius of the circle x² + y² − 4x + 6y − 3 = 0.

General form: 2g=−4→g=−2; 2f=6→f=3; c=−3

Centre = (−g,−f) = (2,−3)

r = √(g²+f²−c) = √(4+9+3) = √16 = 4

✓ Centre = (2,−3), Radius = 4

MediumFind focus and directrix of parabola y²=12x▾

For the parabola y² = 12x, find the focus and directrix.

Compare with y²=4ax: 4a=12 → a=3

Focus = (a,0) = (3,0)

Directrix: x = −a = x = −3

Latus rectum = 4a = 12

✓ Focus = (3,0), Directrix: x = −3

EAPCET LevelFind eccentricity of ellipse 4x²+9y²=36▾

Find the eccentricity of the ellipse 4x² + 9y² = 36.

Divide by 36: x²/9 + y²/4 = 1 → a²=9, b²=4

Since a²>b², major axis is along x-axis.

b² = a²(1−e²) → 4 = 9(1−e²) → 1−e² = 4/9 → e² = 5/9

e = √5/3

✓ Eccentricity = √5/3

Trap QuestionTwo lines are parallel — product of slopes = −1? True or False?▾

Two lines have equations 3x+4y=5 and 6x+8y=10. A student says they're perpendicular since m₁×m₂=−1. Evaluate.

The trap: These equations are MULTIPLES of each other → same line (or parallel), not perpendicular.

Line 1: 3x+4y=5 → m₁ = −3/4

Line 2: 6x+8y=10 → divide by 2 → 3x+4y=5 → same line! m₂ = −3/4

m₁m₂ = (−3/4)(−3/4) = 9/16 ≠ −1

The lines are identical (same slope AND intercept). Perpendicularity requires m₁m₂ = −1.

✓ False — the two equations represent the same line, not perpendicular lines

Mistake DNA

5 coordinate geometry errors from EAPCET distractor analysis.

📐

Perpendicular Slope: Using m instead of −1/m

If a line has slope m, any perpendicular line has slope −1/m (negative reciprocal). Students use the same slope or just −m.

❌ Wrong

Line with slope 2/3: perp slope = −2/3 ✗ (just negated)

✓ Correct

Perp slope = −1/(2/3) = −3/2 ✓ Product m₁m₂ = (2/3)(−3/2) = −1 ✓

Perpendicular lines: m₁ × m₂ = −1 always. Take the negative reciprocal: flip the fraction and change sign.

🔘

Circle General Form: Centre is (g,f), Not (−g,−f)

x²+y²+2gx+2fy+c=0 has centre (−g,−f). The minus signs are frequently missed.

❌ Wrong

x²+y²−4x+6y−3=0: 2g=−4, 2f=6: centre = (−4, 6) ✗ (forgot the negatives)

✓ Correct

g=−2, f=3 centre = (−g,−f) = (2,−3) ✓ Radius = √(g²+f²−c) ✓

In general form x²+y²+2gx+2fy+c=0: compare coefficient of x to 2g, of y to 2f. Then centre = (−g, −f). The negative signs are part of the formula.

🎯

Parabola: Confusing y²=4ax and x²=4ay

y²=4ax opens horizontally (focus on x-axis). x²=4ay opens vertically (focus on y-axis).

❌ Wrong

x²=4ay: focus = (a,0) ✗ (that's for y²=4ax)

✓ Correct

y²=4ax: focus=(a,0) ✓ x²=4ay: focus=(0,a) ✓ Squared variable determines orientation

Which variable is squared tells you the axis of symmetry. y² → parabola symmetric about x-axis. x² → symmetric about y-axis.

⬜

Ellipse: Identifying a and b Correctly

In x²/a²+y²/b²=1, a is associated with x. But 'a' is the LARGER of the two, which may be with y if y-denominator is bigger.

❌ Wrong

x²/4+y²/9=1: a=2 (x-denominator) ✗ (b²=9>4=a², major axis along y)

✓ Correct

a²=max denominator=9 ✓ a=3; b=2 ✓ Major axis along y-axis Foci at (0,±ae) ✓

In x²/p+y²/q=1: if p>q, major axis along x, a²=p, b²=q. If q>p, major axis along y, a²=q, b²=p. Always: a > b.

📏

Area of Triangle: Forgetting the ½ or the Absolute Value

The shoelace formula for triangle area requires ½ AND absolute value. Missing either gives wrong answer.

❌ Wrong

Area = x₁(y₂−y₃)+x₂(y₃−y₁) +x₃(y₁−y₂) ✗ (missing ½ and |..|)

✓ Correct

Area = ½|x₁(y₂−y₃)+x₂(y₃−y₁) +x₃(y₁−y₂)| ✓ Both ½ and |..| needed

The formula gives a signed area. The absolute value ensures a positive result regardless of vertex ordering (clockwise vs anti-clockwise). The ½ converts the parallelogram area to triangle area.

Chapter Intelligence

Coordinate Geometry is the second-highest scoring chapter in EAPCET Maths after Trigonometry.

EAPCET Weightage (2019–2024)

Straight lines (distance, angle, intercepts)

Circles (centre, radius, tangent)

Parabola (focus, vertex, latus rectum)

Ellipse (eccentricity, foci)

Hyperbola (asymptotes, e)

Pair of lines

High-Yield PYQ Patterns

Distance from point to lineCircle centre/radius from general formEquation of tangent to circleParabola focus and directrixEccentricity of ellipse/hyperbolaIntersection of line and circleArea of triangle from verticesPerpendicular bisector of segment

Exam Strategy

For any circle question: convert to general form x²+y²+2gx+2fy+c=0 first, extract g, f, c; then centre=(−g,−f), r=√(g²+f²−c).
Parabola: memorise y²=4ax as the standard (opens right). All four orientations follow from this by sign/axis swaps.
Ellipse: the larger denominator gives a², the smaller gives b². a > b always. Identify major axis orientation first.
Straight lines: the perpendicular distance formula |ax₁+by₁+c|/√(a²+b²) is a direct formula — memorise it for instant MCQ answers.
Coordinate Geometry is linked to Vectors (position vectors of points) and Complex Numbers (Argand plane = 2D coordinate system).

Concept Core

Formula Vault

Worked Examples

Mistake DNA

Chapter Intelligence

💡 Suggestions & Feedback

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