Coordinate Geometry

1. Cartesian System & Shifting of Origin

Fig 1: Visual representation of Cartesian Plane and Conic Sections

The foundation of coordinate geometry lies in the rectangular (Cartesian) coordinate system representing points as (x, y).

Distance Formula: d = √((x₂ - x₁)² + (y₂ - y₁)²)
Section Formula (Internal): x = (mx₂ + nx₁)/(m+n), y = (my₂ + ny₁)/(m+n)

X = x - h, Y = y - k

Shifting of Origin: If the origin is shifted to a new point (h, k) without rotating the axes, the new coordinates (X, Y) of a point originally at (x, y) are given by these relations.

2. Straight Lines

Slope & Angles

The slope of a line is defined as m = tan(θ) = (y₂ - y₁)/(x₂ - x₁).

Angle between two lines: tan(θ) = |(m₁ - m₂) / (1 + m₁m₂)|
Parallel lines: m₁ = m₂
Perpendicular lines: m₁m₂ = -1

Various Forms of Equation

General Form: Ax + By + C = 0
Slope-Intercept Form: y = mx + c
Point-Slope Form: y - y₁ = m(x - x₁)
Intercept Form: x/a + y/b = 1
Normal Form: x cos(α) + y sin(α) = p (where p is perpendicular distance from origin)

d = |Ax₁ + By₁ + C| / √(A² + B²)

Perpendicular distance of a point (x₁, y₁) from the line Ax + By + C = 0.

3. Conic Sections: Fundamentals

A conic section is the locus of a point whose distance from a fixed point (Focus) bears a constant ratio, e (eccentricity), to its perpendicular distance from a fixed straight line (Directrix).

Circle: e = 0
Parabola: e = 1
Ellipse: 0 < e < 1
Hyperbola: e > 1

General 2nd Degree Equation & Degenerate Cases

Equation: ax² + 2hxy + by² + 2gx + 2fy + c = 0

Let discriminant Δ = abc + 2fgh - af² - bg² - ch².

Δ = 0 (Degenerate Conics):
Degenerate Conics (Visual Meaning)
- Pair of straight lines → represents two intersecting lines
- Parallel lines → no intersection
- Single line → repeated root case
- Point → collapsed conic
Δ ≠ 0 (Non-degenerate Conics):
- h² = ab &implies; Parabola
- h² < ab &implies; Ellipse
- h² > ab &implies; Hyperbola

4. Circles

(x - h)² + (y - k)² = r²

Standard equation of a circle with center (h, k) and radius r.

General Equation

x² + y² + 2gx + 2fy + c = 0

Center: (-g, -f)
Radius: √(g² + f² - c)
Condition to be a real circle: g² + f² - c ≥ 0.

Solved Example: Circle

Q: Find the center and radius of x² + y² - 4x + 6y - 12 = 0.

Solution: Here 2g = -4 ⇒ g = -2, 2f = 6 ⇒ f = 3, c = -12.
Center: (-g, -f) = (2, -3).
Radius: √((-2)² + 3² - (-12)) = √(4 + 9 + 12) = √25 = 5 units.

5. Parabola

Standard Equation: y² = 4ax

Vertex: (0, 0)
Focus: (a, 0)
Directrix: x = -a
Length of Latus Rectum: 4a

Other Standard Forms

x² = 4ay → Opens upward
x² = -4ay → Opens downward
y² = -4ax → Opens left

IAT Shortcut: Parametric Form

Any point on the parabola y² = 4ax can be taken as (at², 2at). This greatly simplifies locus problems!

6. Ellipse & Hyperbola

Ellipse: `x²/a² + y²/b² = 1` (a > b)

Foci: (±ae, 0)
Directrices: x = ±a/e
Eccentricity relation: b² = a²(1 - e²)
Latus Rectum: 2b²/a

Hyperbola: `x²/a² - y²/b² = 1`

Transverse & Conjugate Axes: Lengths 2a and 2b respectively.
Foci: (±ae, 0)
Directrices: x = ±a/e
Eccentricity relation: b² = a²(e² - 1)
Latus Rectum: 2b²/a

y = ± (b/a)x

Asymptotes of Hyperbola: For x²/a² - y²/b² = 1.

7. IAT Exam Focus & Magic Rules

The "T = 0" Magic: To find the equation of a tangent at point (x₁, y₁) on any conic, replace: x² → xx₁, y² → yy₁, x → (x+x₁)/2, and y → (y+y₁)/2.
Position of a Point: Plug the point (x₁, y₁) into the equation (say S = 0). If S₁ < 0, it's inside the curve. If S₁ > 0, it's outside.
Director Circle: Locus of intersection of perpendicular tangents. For circle: x²+y²=2r². For ellipse: x²+y²=a²+b².

8. Practice Mock Test

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