Analytic Geometry
Analytic geometry connects algebra and shape. Bagrut questions cover lines, circles, distance and midpoint, perpendicularity, intersections, and loci, often asking for an equation that satisfies given geometric conditions.
Core rules
- Distance: d=√[(x₂−x₁)²+(y₂−y₁)²]. Midpoint: ((x₁+x₂)/2,(y₁+y₂)/2).
- Slope: m=(y₂−y₁)/(x₂−x₁). Parallel lines have equal slopes; perpendicular lines have m₁m₂=−1.
- Line forms: y=mx+b and y−y₁=m(x−x₁).
- Circle: (x−a)²+(y−b)²=r² has centre (a,b) and radius r.
- A tangent to a circle is perpendicular to the radius at the point of contact.
- Distance from (x₀,y₀) to Ax+By+C=0 is |Ax₀+By₀+C|/√(A²+B²).
Worked example
Find the centre and radius of x²+y²−4x+6y−12=0. Complete the square: x²−4x=(x−2)²−4 and y²+6y=(y+3)²−9. Substitute: (x−2)²−4+(y+3)²−9−12=0, so (x−2)²+(y+3)²=25. The centre is (2,−3) and the radius is 5. Completing the square avoids sign errors in reading the centre.
Traps
Reading the centre as (−2,3) instead of (2,−3) is a classic sign error. A zero denominator means a vertical line, not an infinite slope to ignore. When finding intersections, solve the system and check both variables. Do not assume a line is tangent without verifying the distance from the centre equals the radius.