Coordinate Geometry: Locus & Transformation

Locus is the set of all points satisfying a given geometric condition, expressed as an algebraic equation in coordinates. Coordinate transformations -- translation, rotation, and reflection -- change the frame of reference to simplify equations or reveal geometric properties. Together, locus and transformation form the foundational language of coordinate geometry.

Locus Definition and Procedure: To find a locus: (1) Let P(h,k) be a general point on the locus. (2) Express the given geometric condition as an equation involving h, k and known quantities. (3) Eliminate any auxiliary parameters to get a relation purely in h and k. (4) Replace h by x and k by y to state the locus equation.

Common Locus Problems: The locus of a point equidistant from two fixed points is the perpendicular bisector of the segment joining them. The locus equidistant from a point and a line is a parabola. The locus where the sum of distances from two fixed points is constant is an ellipse, and the difference gives a hyperbola.

Translation of Axes: Shifting the origin to (h,k): new coordinates X = x - h, Y = y - k. The equation f(x,y) = 0 becomes f(X+h, Y+k) = 0. Translation does not change the shape or size of curves, only the reference point. It is used to remove first-degree terms from a conic equation.

Rotation of Axes: Rotating axes by angle $\theta$: x = Xcos($\theta$) - Ysin($\theta$), y = Xsin($\theta$) + Ycos($\theta$).
Equivalently, X = xcos($\theta$) + ysin($\theta$), Y = -xsin($\theta$) + ycos($\theta$).
Rotation is used to eliminate the xy-term in the general second-degree equation $ax^{2}$ + 2hxy + $by^{2}$ + 2gx + 2fy + c = 0.
The required angle satisfies tan(2*$\theta$) = 2h/(a-b).

Reflection: Reflection across x-axis: (x,y) → (x,-y). Across y-axis: (x,y) → (-x,y). Across origin: (x,y) → (-x,-y). Across y = x: (x,y) → (y,x). Across y = -x: (x,y) → (-y,-x). Across a general line y = mx + c: use the perpendicular-foot-image formula.

Invariants Under Transformation: The discriminant $\Delta$ = abc + 2fgh - $af^{2}$ - $bg^{2}$ - $ch^{2}$ of the general conic is invariant under rotation and translation. The quantities a+b and $h^{2}$-ab are also invariant. These determine the conic type regardless of coordinate choice.

The key problem-solving concept is systematically eliminating parameters from the geometric condition to obtain the locus equation, and choosing the right transformation to simplify a given conic equation.

---