A7LAM CSCA ACADEMYCSCA Academy / Mathematics / Geometry and algebra
CSCA Mathematics · Lesson 10 of 16Lines and circles
Droites et cercles · المستقيمات والدوائر
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Analytic geometry turns shapes into equations. For a line you need its slope; for a circle, its centre and radius.
Key ideas
- Line: y = kx + b (slope k), or Ax + By + C = 0.
- Parallel lines have equal slopes; perpendicular slopes multiply to −1.
- Circle: (x − a)² + (y − b)² = r², centre (a, b), radius r.
- A line is tangent to a circle when the distance from the centre to the line equals r.
Formulas to know
- Point to line: d = |Ax₀ + By₀ + C| / √(A² + B²)
- Distance: √((x₂ − x₁)² + (y₂ − y₁)²)
- Midpoint: ((x₁ + x₂)/2, (y₁ + y₂)/2)
Worked example
Find the centre and radius of x² + y² − 4x + 2y − 4 = 0.
- Group: (x² − 4x) + (y² + 2y) = 4
- Complete: (x − 2)² + (y + 1)² = 4 + 4 + 1 = 9
- Read: centre (2, −1), r = √9 = 3
Answer: Centre (2, −1), radius 3
Practice questions
1. Slope of the line through (1, 2) and (3, 8)?
2 · 3 ✓ · 6 · 1/3
(8 − 2)/(3 − 1) = 3.
2. Which line is perpendicular to y = 2x + 1?
y = 2x − 3 · y = −2x · y = −½x + 4 ✓ · y = ½x
2 × (−½) = −1.
3. Distance from (0, 0) to the line 3x + 4y − 10 = 0?
2 ✓ · 10 · 5 · 2.5
|−10| / √(9 + 16) = 10/5 = 2.