Analytic Geometry
Core concepts
Know the distance formula, midpoint formula, slope, and equations of lines (slope-intercept, point-slope, general form). Master the equation of a circle (x − a)² + (y − b)² = r², the conditions for parallel (equal slopes) and perpendicular (product = −1) lines, and the distance from a point to a line. Recognize the parabola y = ax² + bx + c with vertex and axis of symmetry, and the standard forms of ellipse and hyperbola at a basic level.
Traps
- Forgetting that vertical lines have undefined slope, not zero slope.
- Sign errors in the circle equation when the center has negative coordinates.
- Using the wrong sign in the perpendicular slope (it is the negative reciprocal).
- Confusing the distance from a point to a line with the distance between two points.
- Misidentifying which conic a general second-degree equation represents without checking coefficients.
Worked example
Find the equation of the line through (2, −1) perpendicular to 3x − 4y = 5. The given line has slope 3/4, so the perpendicular slope is −4/3. Using point-slope: y + 1 = (−4/3)(x − 2), giving 4x + 3y = 5.
Rule of thumb
For perpendicular lines, flip the slope and change its sign; for parallel lines, keep the slope identical and adjust only the intercept.