Definitions [10]
The equation of the locus of a point is the algebraic relation which is satisfied by the coordinates of every point on the locus of the point.
Locus is the path traced by a moving point, which moves so as to satisfy a certain given condition/conditions.
An equation of the form ax + by + c = 0 represents a straight line and is known as a linear equation.
The straight line passing through the focus and perpendicular to the directrix is called the axis of the conic section.
The points of intersection of the conic section and the axis are called the vertices of the conic section.
The chord passing through the focus and perpendicular to the axis is called the latus rectum of the conic section.
A chord of a conic passing through the focus is called a focal chord.
A straight line drawn perpendicular to the axis and terminating at both ends of the curve is a double ordinate of the conic section.
A conic section is the locus of a point such that the ratio of its distance from a fixed point (focus) to a fixed line (directrix) is constant.
The point which bisects every chord of the conic passing through it is called the centre of the conic section.
Formulae [2]
From general form:
- Slope (m) = −a / b
- Y-intercept = −c / b
$$e = \frac{\text{distance from focus}}{\text{distance from directrix}}$$
Key Points
- Step I: Take any point P(x, y) on the locus.
- Step II: Write down the geometrical condition of the locus.
- Step III: Convert the geometrical condition into an algebraic equation involving x and y.
- Step IV: Simplify the equation to get the required “equation of the locus”.
| Form | Formula |
|---|---|
| X-axis | y = 0 |
| Y-axis | x = 0 |
| Parallel to the X-axis | y = b or y = -b |
| Parallel to the Y-axis | x = a or x = -a |
| Slope-point form | y − y₁ = m(x − x₁) |
| Two-point form | \[\frac{y-y_{1}}{y_{1}-y_{2}}=\frac{x-x_{1}}{x_{1}-x_{2}}\] |
| Slope-intercept form | y = mx + c |
| Intercept form | \[\frac{x}{\mathrm{a}}+\frac{y}{\mathrm{b}}=1\] |
| Normal form | x cosα + y sinα = p |
| Parametric form | \[\frac{x-x_{1}}{\cos\theta}=\frac{y-y_{1}}{\sin\theta}=r\] |
Position of a Point:
For line: ax₁ + by₁ + c
- If ax₁ + by₁ + c = 0 → Point lies on the line
- If ax₁ + by₁ + c < 0 → Point lies on one side (origin side)
- If ax₁ + by₁ + c > 0 → Point lies on other side
