Coordinate Geometry

🟢 Lite — Quick Review

Rapid summary for last-minute revision before your exam.

Coordinate Geometry — Key Facts for Kenyatta University Distance: d = √[(x₂−x₁)² + (y₂−y₁)²] Straight line: y = mx + c; slope m = (y₂−y₁)/(x₂−x₁) Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2) Circle: (x−h)² + (y−k)² = r²; centre (h,k), radius r ⚡ Exam tip: Kenyatta coordinate geometry focuses on straight lines and circles — 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 — Kenyatta University Study Guide

Distance formula: d = √[(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)| Collinear if 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 form: x/a + y/b = 1

Angle between lines: 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)

Circle:

• Standard: (x−h)² + (y−k)² = r²
• General: x² + y² + 2gx + 2fy + c = 0; centre (−g, −f), radius = √(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

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

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

• Key formula: Distance: √[(Δx)²+(Δy)²]; Circle: centre (−g,−f), radius √(g²+f²−c)
• Common trap: For circle general form, radius = √(g²+f²−c), not just √(g²+f²)
• Exam weight: 2–3 questions per exam

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🔴 Extended — Deep Dive

Comprehensive coverage for students on a longer study timeline.

Coordinate Geometry — Comprehensive Kenyatta Notes

Family of lines:

• Through intersection: L₁ + λL₂ = 0
• Through (x₁,y₁): y − y₁ = m(x − x₁)

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

Angle bisectors: (L₁)/√(A₁²+B₁²) = ± (L₂)/√(A₂²+B₂²)

Conic identification: Using discriminant B² − AC:

• < 0: ellipse
• = 0: parabola

• 0: hyperbola

Chord of circle: Chord with midpoint (x₁,y₁): T = S₁

Director circle: For ellipse x²/a² + y²/b² = 1: x² + y² = a² + b²

Parametric forms:

• Circle: x = h + r cos θ, y = k + r sin θ
• Ellipse: x = a cos θ, y = b sin θ
• Parabola: x = at², y = 2at

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

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

Locus problems:

1. Express condition algebraically
2. Eliminate parameter
3. Get equation in x, y

Rotation of axes: tan 2θ = 2B/(A−C) to remove xy term

Shift of origin: Complete squares in x and y to simplify

Polar coordinates: x = r cos θ, y = r sin θ r² = x² + y², tan θ = y/x

Distance from point to line in polar: Uses formula conversion

Parametric line: (x−x₁)/a = (y−y₁)/b = (z−z₁)/c (3D)

3D distance: d = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]

Plane in 3D: Ax + By + Cz + D = 0; normal vector = (A,B,C)

Distance from point to plane: d = |Ax₁ + By₁ + Cz₁ + D|/√(A²+B²+C²)

• Remember: Circle centre (−g,−f), radius √(g²+f²−c); parallel lines: m₁ = m₂; perpendicular: m₁m₂ = −1
• Previous years: “Find equation of circle through (1,2) and (3,4) with centre on x-axis” [2023 KU]; “Find distance between parallel lines 2x + 3y = 5 and 4x + 6y = 10” [2024 KU]; “Find focus of parabola y² = 8x” [2024 KU]

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📊 Kenyatta University Exam Essentials

Detail	Value
Questions	50 (depending on course)
Time	2–3 hours
Marks	100
Format	Mix of short answer and problem solving

💡 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 locus problems, eliminate parameter by expressing condition algebraically
• For parallel/perpendicular lines, use slope relationship

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