Definitions [6]
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.
The slope m of a line is m = tanθ
where θ is the inclination of the line with the positive x-axis.
An equation of the form ax + by + c = 0 represents a straight line and is known as a linear equation.
Three or more lines are concurrent if they meet at a single point.
If L₁: a₁x + b₁y + c₁ = 0 and L₂: a₂x + b₂y + c₂ = 0 represent two intersecting lines, then equation L₁ + λL₂ = 0, λ ∈ R, represents a family of lines.
Formulae [9]
| Sr. No. | Name | Condition | Formula |
|---|---|---|---|
| i. | Distance Formula | Two points P(x₁, y₁), Q(x₂, y₂) | \[\mathrm{d(PQ)}=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\] |
| a. | Internal Division | P divides AB in the ratio m: n | \[\mathrm{P\equiv\left(\frac{mx_{2}+nx_{1}}{m+n},\frac{my_{2}+ny_{1}}{m+n}\right)}\] |
| b. | Midpoint Formula | P is the midpoint of AB | \[\mathrm{P}\equiv\left(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2}\right)\] |
| c. | External Division | P divides AB externally in m: n | \[\mathrm{P\equiv\left(\frac{mx_{2}-nx_{1}}{m-n},\frac{my_{2}-ny_{1}}{m-n}\right)}\] |
| iii. | Centroid Formula | Triangle with vertices A, B, C | \[\left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}\right)\] |
If the origin is shifted from O(0, 0) to O′(h, k), then:
- x = X + h
- y = Y + k
\[m=\frac{y_2-y_1}{x_2-x_1}\]
From general form:
- Slope (m) = −a / b
- Y-intercept = −c / b
If slopes are m1 and m2:
\[\tan\theta=\left|\frac{m_1-m_2}{1+m_1m_2}\right|\]
If lines are a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0,
\[\tan\theta=\left|\frac{a_1b_2-a_2b_1}{a_1a_2+b_1b_2}\right|\]
Conditions for Parallel, Perpendicular and Identical Lines:
Parallel Lines:
Slope: m₁ = m₂
In general form: \[\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}\]
Perpendicular Lines:
Slope: m₁m₂ = −1
In general form: a₁a₂ + b₁b₂ = 0
Identical Lines:
\[\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\]
For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0,
\[(x,y)=\left(\frac{b_1c_2-b_2c_1}{a_1b_2-a_2b_1},\frac{c_1a_2-c_2a_1}{a_1b_2-a_2b_1}\right)\]
If ax² + 2hxy + by² + 2gx + 2fy + c = 0 represents a pair of parallel straight lines, then the distance between them is given by
\[2\sqrt{\frac{g^{2}-ac}{a(a+b)}}\mathrm{or}2\sqrt{\frac{f^{2}-bc}{b(a+b)}}\]
For point (x₁, y₁) and line ax + by + c = 0,
\[p=\frac{|ax_1+by_1+c|}{\sqrt{a^2+b^2}}\]
For lines ax + by + c₁ = 0 and ax + by + c₂ = 0,
P = \[\left|\frac{c_{1}-c_{2}}{\sqrt{a^{2}+b^{2}}}\right|\]
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”.
Nature of Slope
-
m > 0 → rising line
-
m < 0 → falling line
-
m = 0 → horizontal line
-
m = ∞→ vertical line
Parallel Lines
Two lines are parallel ⇔ , their slopes are equal, m1 = m2
Perpendicular Lines
Two lines are perpendicular ⇔
Collinearity of Three Points
Points A, B, and C are collinear
Method 1: Distance method
AB + BC = AC
Method 2: Slope method
Slope of AB = Slope of BC
| 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
Condition for concurrency:
For lines
a₁x + b₁y + c₁ = 0
a₂x + b₂y + c₂ = 0
a₃x + b₃y + c₃ = 0
\[\begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}=0\]
For line: ax + by + c = 0
- If ax₁ + by₁ + c and c have the same sign → point lies on the same side as the origin
- If ax₁ + by₁ + c and c have opposite sign → point lies on opposite side
