Extended Concept (LE)

📐 Coordinate Geometry: Concepts & Formulae

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1. Distance Formula

Concept:
The distance formula is used to find the length of the line segment joining two points in a plane. If the coordinates of the two points are \((x_1, y_1)\) and \((x_2, y_2)\), the distance between them is calculated using the Pythagorean theorem.

Formula:

$$ d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2} $$

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2. Section Formula

Concept:
The section formula is used to find the coordinates of a point which divides the line joining two given points in a certain ratio.

(i) Mid-Point Formula

Concept:
The mid-point is the point which divides the line segment joining two points into two equal parts.

Formula:

$$ (x, y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) $$

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(ii) Ratio Formula

Concept:
If a point divides the line joining two points in the ratio \(m_1 : m_2\), then the coordinates of that point can be found using the ratio formula.

Formula:

$$ (x, y) = \left( \frac{m_1x_2 + m_2x_1}{m_1 + m_2}, \frac{m_1y_2 + m_2y_1}{m_1 + m_2} \right) $$

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(iii) Centroid Formula

Concept:
The centroid of a triangle is the point where all three medians intersect. If the vertices of the triangle are \((x_1,y_1)\), \((x_2,y_2)\), and \((x_3,y_3)\), then the centroid is the average of their coordinates.

Formula:

$$ (x, y) = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) $$

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3. Intercepts of a Line

Concept:
To draw the graph of a straight line, first put \(y = 0\) to find the x-intercept, then put \(x = 0\) to find the y-intercept. Joining these two points gives the required graph.

Intercept Formula for the General Equation:

$$ ax + by + c = 0 $$

• X-intercept: $$ x = -\frac{c}{a} $$
• Y-intercept: $$ y = -\frac{c}{b} $$

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4. Important Remarks

• Linear relationship means the rate of change is constant.
• Slope represents the rate of change.
• Y-intercept represents the initial value.
• X-intercept is the point where the graph touches the x-axis.
• Y-intercept is the point where the graph touches the y-axis.
• Time is always taken on the horizontal (x-axis).
• Frequency or output is taken on the vertical (y-axis).

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5. Equation of Straight Lines

(i) Slope-Intercept Form

$$ y = mx + c $$

(ii) Point-Slope Form

$$ (y – y_1) = m(x – x_1) $$

(iii) Two-Point Form

$$ (y – y_1) = \frac{y_2 – y_1}{x_2 – x_1}(x – x_1) $$

(iv) Double Intercept Form

$$ \frac{x}{a} + \frac{y}{b} = 1 $$

(v) Line Parallel to X-axis

$$ y = b \quad (b = \text{constant}) $$

(vi) Line Parallel to Y-axis

$$ x = a \quad (a = \text{constant}) $$

(vii) Equations of Coordinate Axes

$$ x\text{-axis: } y = 0 \qquad y\text{-axis: } x = 0 $$