Coordinate Geometry

🟢 Lite — Quick Review

Rapid summary for last-minute revision before your exam.

Coordinate Geometry — Key Facts for BUET Straight line: y = mx + c; slope m = (y₂−y₁)/(x₂−x₁); equation through (x₁,y₁): y−y₁ = m(x−x₁) Circle: (x−h)² + (y−k)² = r²; general: x² + y² + 2gx + 2fy + c = 0 with centre (−g, −f), radius = √(g²+f²−c) Distance point to line: |Ax₁ + By₁ + C|/√(A²+B²) ⚡ Exam tip: BUET coordinate geometry problems often involve circles and tangents — T = 0 for tangent at point on circle is essential!

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🟡 Standard — Core Study

Standard content for students with a few days to months.

Coordinate Geometry — BUET Study Guide

Distance formula: Distance between (x₁, y₁) and (x₂, y₂) = √[(x₂−x₁)² + (y₂−y₁)²]

Section formula: Point dividing P₁P₂ in ratio m:n (internal): [(mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)] Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2)

Area of triangle: Area = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)| Three points are collinear if this area = 0

Straight lines:

• Slope-intercept: y = mx + c
• Point-slope: y − y₁ = m(x − x₁)
• Two-point: (y−y₁)/(y₂−y₁) = (x−x₁)/(x₂−x₁)
• Intercept: x/a + y/b = 1
• Normal form: x cos α + y sin α = p

Angle between two lines: If slopes m₁, m₂: tan θ = |(m₂ − m₁)/(1 + m₁m₂)| Parallel: m₁ = m₂; Perpendicular: m₁m₂ = −1

Perpendicular distance: From point (x₁, y₁) to line Ax + By + C = 0: d = |Ax₁ + By₁ + C|/√(A²+B²) Between parallel lines Ax + By + C₁ = 0 and Ax + By + C₂ = 0: d = |C₁ − C₂|/√(A²+B²)

Circle:

• Standard: (x−h)² + (y−k)² = r², centre (h, k), radius r
• General: x² + y² + 2gx + 2fy + c = 0, centre (−g, −f), radius = √(g²+f²−c)
• Condition for real circle: g² + f² > c

Tangent to circle: At point (x₁, y₁) on x² + y² + 2gx + 2fy + c = 0: use T = 0 xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0

Length of tangent: From point (x₁, y₁) to circle: √[S₁₁] where S₁₁ = x₁² + y₁² + 2gx₁ + 2fy₁ + c

Parabola: y² = 4ax; focus (a, 0); directrix x = −a; vertex (0, 0) Parametric: (at², 2at)

Ellipse: x²/a² + y²/b² = 1 (a > b); focus (±c, 0) where c² = a² − b²; eccentricity e = c/a < 1

Hyperbola: x²/a² − y²/b² = 1; focus (±c, 0) where c² = a² + b²; e = c/a > 1

• Key formula: Distance point-line: |Ax₁+By₁+C|/√(A²+B²); Circle centre: (−g, −f); tan θ = |(m₂−m₁)/(1+m₁m₂)|
• Common trap: For circle in general form, radius = √(g²+f²−c), not √(g²+f²)
• Exam weight: 2–3 questions per exam (8–12 marks); combined with calculus often

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🔴 Extended — Deep Dive

Comprehensive coverage for students on a longer study timeline.

Coordinate Geometry — Comprehensive BUET Notes

Family of lines:

• Through intersection of L₁ = 0 and L₂ = 0: L₁ + λL₂ = 0
• Through (x₁, y₁): y − y₁ = m(x − x₁) for any slope m
• Parallel to Ax + By + C = 0: Ax + By + k = 0
• Perpendicular to Ax + By + C = 0: Bx − Ay + k = 0

Pair of straight lines: ax² + 2hxy + by² = 0 represents two lines through origin Angle between them: tan θ = 2√(h²−ab)/(a+b)

Angle bisectors: For lines L₁ = 0 and L₂ = 0, bisectors are given by: (L₁)/√(A₁²+B₁²) = ± (L₂)/√(A₂²+B₂²)

• gives bisector of angle containing origin (if constants positive), or the acute angle bisector

Rotation of axes: For removing xy term: tan 2θ = 2B/(A−C) where original is Ax² + 2Bxy + Cy² + … = 0

Conic identification: Using discriminant B² − AC:

• B² − AC < 0: ellipse (or circle if A = C and B = 0)
• B² − AC = 0: parabola
• B² − AC > 0: hyperbola

Chord of circle: Chord with midpoint (x₁, y₁) for x² + y² + 2gx + 2fy + c = 0: T = S₁

Chord of contact: From external point (x₁, y₁): use T = 0

Director circle: For ellipse x²/a² + y²/b² = 1: x² + y² = a² + b² Locus of point from which pair of perpendicular tangents can be drawn

Parametric circle: x = h + r cos θ, y = k + r sin θ

Parametric ellipse: x = a cos θ, y = b sin θ

Parametric parabola: x = at², y = 2at

Tangent to parabola y² = 4ax: At (at², 2at): ty = x + at²

Tangent to ellipse x²/a² + y²/b² = 1: At (a cos θ, b sin θ): (x cos θ)/a + (y sin θ)/b = 1

Asymptotes of hyperbola: x²/a² − y²/b² = 1: asymptotes are y = ±(b/a)x

Polar equation of line: r = −e cos(θ − α)/[cos(θ − φ)] … different forms available

Condition for tangency:

• Parabola y² = 4ax: line y = mx + c touches if c = a/m
• Ellipse x²/a² + y²/b² = 1: line y = mx + c touches if c² = a²m² + b²
• Hyperbola x²/a² − y²/b² = 1: line y = mx + c touches if c² = a²m² − b²

Locus problems:

1. Express condition in coordinates
2. Eliminate parameter to get equation in x, y only

Shift of origin: For removing linear terms, complete the square in x and y separately

• Remember: Circle centre (−g, −f), radius √(g²+f²−c); parallel: m₁ = m₂; perpendicular: m₁m₂ = −1; for conics: discriminant B²−AC identifies type
• Previous years: “Find equation of circle through (1,2), (3,4) with centre on x-axis” [2023 BUET]; “Find distance between parallel lines 2x + 3y = 5 and 4x + 6y = 8” [2024 BUET]; “Find focus and directrix of y² = 8x” [2024 BUET]

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📊 BUET Admission Exam Essentials

Detail	Value
Questions	Varies by year (~40-50 MCQ)
Time	Usually 2–3 hours
Marks	Varies by section
Subjects	Mathematics (highest weight), Physics, Chemistry
Negative	Usually no negative marking in BUET
Mode	Written + MCQ depending on year

🎯 High-Yield Topics for BUET Mathematics

• Calculus (Differentiation + Integration) — highest weight
• Algebra (Quadratics, AP/GP/HP) — very high weight
• Coordinate Geometry (Circle, Conics) — high weight
• Trigonometry — medium-high weight
• Complex Numbers — medium weight

📝 Previous Year Question Patterns

• Coordinate Geometry: 2–3 questions per exam, 8–12 marks
• Common patterns: circle equations, tangent at point, distance point-line, locus
• Weight: high — frequently combined with calculus

💡 Pro Tips

• For circle problems, always find centre and radius first
• T = 0 for tangent at point on circle; S₁ = 0 for point on circle condition
• For family of lines through intersection, use L₁ + λL₂ = 0
• When checking number of common tangents between circles, first find distance between centres
• Locus problems: eliminate parameter by expressing condition algebraically, then solve

🔗 Official Resources

• BUET Official
• BUET Admission Portal

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