Unit 8: Coordinate Geometry

Coordinate geometry (or analytic geometry) is defined as the study of geometry using coordinate systems. It bridges algebra and geometry, allowing geometric shapes to be defined by algebraic equations.

1. Straight Line

A straight line is a set of points extending infinitely in opposite directions.

Gradient (Slope) of a Line

The gradient $m$ of a line passing through $(x_1, y_1)$ and $(x_2, y_2)$ is: $$ m = \frac{y_2 - y_1}{x_2 - x_1} \quad (x_1 \neq x_2) $$

Equations of a Line

1. Parallel to Axes: $x = a$ (parallel to y-axis), $y = b$ (parallel to x-axis)
2. Point-Slope Form: $y - y_1 = m(x - x_1)$
3. Two-Point Form: $y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$
4. Slope-Intercept Form: $y = mx + c$
5. Intercept Form: $\frac{x}{a} + \frac{y}{b} = 1$

Distance from a Point to a Line

The perpendicular distance from $(x_1, y_1)$ to the line $ax + by + c = 0$ is: $$ d = \frac{|a x_1 + b y_1 + c|}{\sqrt{a^2 + b^2}} $$

Distance between Parallel Lines

For $ax + by + c_1 = 0$ and $ax + by + c_2 = 0$: $$ d = \frac{|c_1 - c_2|}{\sqrt{a^2 + b^2}} $$

2. Circle

A circle is the locus of a point moving in a plane such that its distance from a fixed point (center) is constant.

Equations of a Circle

1. Standard Form (Center at origin): $x^2 + y^2 = r^2$
2. Central Form (Center at $(h, k)$): $(x - h)^2 + (y - k)^2 = r^2$
3. Diameter Form: If $(x_1,y_1)$ and $(x_2,y_2)$ are endpoints of a diameter: $$ (x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0 $$
4. General Form: $x^2 + y^2 + 2gx + 2fy + c = 0$
   • Center: $(-g, -f)$
   • Radius: $r = \sqrt{g^2 + f^2 - c}$

3. Parabola

A parabola is the locus of a point whose distance from a fixed point (focus) is equal to its perpendicular distance from a fixed straight line (directrix).

Standard Equations of a Parabola

1. Rightward Opening: $y^2 = 4ax$
   • Focus: $(a, 0)$
   • Directrix: $x = -a$
   • Latus Rectum Length: $4a$
2. Upward Opening: $x^2 = 4ay$
   • Focus: $(0, a)$
   • Directrix: $y = -a$

Competency-Based Questions

1. (Application of Straight Lines): An architect is modeling a roof slope linearly using a CAD software. The highest point is at coordinate $(15, 20)$ and the gutter starts at $(5, 8)$. What is the gradient of the roof, and what is its linear equation in slope-intercept form?

2. (Distance Optimization): A new fiber-optic cable is plotted along the line $3x - 4y + 12 = 0$. A server room is located at map coordinates $(2, -1)$. What is the shortest (perpendicular) length of wire required to connect the server room to the fiber route?

3. (Circle Area Analysis): A cell tower’s coverage area on a digital map is given by the equation $x^2 + y^2 - 6x + 8y - 75 = 0$. Determine the coordinates of the cell tower (the center) and the coverage radius.

4. (Parabolic Antennas): Satellite TV dishes are parabolic. If the cross-section of a dish is modeled by the equation $y^2 = 32x$ (measurements in cm), where should the receiver (focus) be placed relative to the vertex to capture the strongest signal?

5. (Economics Demand Curve): A linear demand curve relates price ($y$) and quantity demanded ($x$). At a price of ₹50, 100 units are demanded. At a price of ₹40, 120 units are demanded. Use the Two-Point Form of a straight line to find the demand equation, and predict the demand at a price of ₹25.

Answers to Competency-Based Questions

1. Application of Straight Lines: Gradient $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{20 - 8}{15 - 5} = \frac{12}{10} = 1.2$. Using point-slope form with point $(5, 8)$: $$ y - 8 = 1.2(x - 5) \implies y = 1.2x - 6 + 8 $$ Linear Equation: $y = 1.2x + 2$.

2. Distance Optimization: Distance $d$ from point $(2, -1)$ to line $3x - 4y + 12 = 0$: $$ d = \frac{|3(2) - 4(-1) + 12|}{\sqrt{3^2 + (-4)^2}} = \frac{|6 + 4 + 12|}{\sqrt{9 + 16}} = \frac{22}{\sqrt{25}} = \frac{22}{5} $$ The shortest length of wire is $4.4$ units.

3. Circle Area Analysis: Comparing $x^2 + y^2 - 6x + 8y - 75 = 0$ with the general form $x^2 + y^2 + 2gx + 2fy + c = 0$:

• $2g = -6 \implies g = -3$
• $2f = 8 \implies f = 4$
• $c = -75$ Center of Tower $(-g, -f) = (3, -4)$. Radius $r = \sqrt{g^2 + f^2 - c} = \sqrt{(-3)^2 + 4^2 - (-75)} = \sqrt{9 + 16 + 75} = \sqrt{100} = \textbf{10 units}$.

4. Parabolic Antennas: The equation is $y^2 = 32x$. This maps to the right-opening parabola standard form $y^2 = 4ax$. $$ 4a = 32 \implies a = 8 $$ The receiver (focus) must be placed at $(a, 0)$, which is $8$ cm directly in front of the vertex of the dish.

5. Economics Demand Curve: Let $(x, y)$ be $(\text{demand}, \text{price})$. Points are $(100, 50)$ and $(120, 40)$. Two-Point Form: $$ y - 50 = \frac{40 - 50}{120 - 100}(x - 100) $$ $$ y - 50 = \frac{-10}{20}(x - 100) = -0.5(x - 100) $$ Demand Equation: $y = -0.5x + 100$. Rearranged for Quantity Given Price ($x$): $$ 0.5x = 100 - y \implies x = 200 - 2y $$ At a price of $y = ₹25$: $$ x = 200 - 2(25) = 200 - 50 = \textbf{150 units demanded} $$