Coordinate Geometry: Straight Lines
What the exam tests
This domain tests slope, equations of lines in different forms, parallelism and perpendicularity, intersection, and distance from a point to a line. Business applications include cost and revenue lines, break-even points and linear demand.
Core rules
- Slope through ((x_1,y_1),(x_2,y_2)) is (m=\frac{y_2-y_1}{x_2-x_1}), provided (x_2\ne x_1).
- Point-slope form: (y-y_1=m(x-x_1)). Slope-intercept: (y=mx+c). Two-point form follows from the slope formula.
- For (Ax+By+C=0), slope is (-A/B) if (B\ne0).
- Parallel lines have equal slopes; perpendicular lines satisfy (m_1m_2=-1).
- Distance from ((x_0,y_0)) to (Ax+By+C=0) is (\frac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}}).
Worked example
Find the line through ((2,3)) perpendicular to (3x+4y=7). The given line has slope (-3/4), so the required slope is (4/3). Using point-slope form: (y-3=\frac43(x-2)). Multiplying by 3 gives (3y-9=4x-8), so (4x-3y+1=0). Substituting ((2,3)) gives (8-9+1=0), confirming the point lies on the line.
Traps
A vertical line has undefined slope, so do not force it into (y=mx+c). The perpendicular condition is a product of (-1), not equality. Keep the signs in point-slope form. In the distance formula, the denominator uses the coefficients (A,B), not the point coordinates. When finding an intercept, set the other variable to zero.