Unit 1: Transformation of Coordinates and Pair of Straight Lines

Transformation of Coordinates

Transformation of coordinates involves changing the coordinate system (the axes and/or origin) to simplify the equation of a curve.

Change of Origin (Translation of Axes)

This transformation involves shifting the origin (0, 0) to a new point (h, k) without changing the direction (orientation) of the axes. The new axes remain parallel to the old axes.

Let (x, y) be the coordinates of a point P with respect to the old axes (OX, OY), and let (x', y') be the coordinates of the same point P with respect to the new axes (O'X', O'Y').

The transformation formulas are:

x = x' + h
y = y' + k

Or, rearranging for the new coordinates:

x' = x - h
y' = y - k

Example:

Question: Transform the equation x² + y² - 4x + 6y - 3 = 0 by shifting the origin to the point (2, -3).

Solution:
Here, h = 2 and k = -3.
Substitute x = x' + 2 and y = y' - 3 into the equation:

(x' + 2)² + (y' - 3)² - 4(x' + 2) + 6(y' - 3) - 3 = 0
(x'² + 4x' + 4) + (y'² - 6y' + 9) - 4x' - 8 + 6y' - 18 - 3 = 0

Now, group the terms:

The new, simplified equation is a circle centered at the new origin.

Rotation of Axes (Orthogonal Transformation)

This transformation involves rotating the axes (OX, OY) by an angle θ about the origin, without changing the origin itself. This is an **orthogonal transformation** because it preserves distances and angles.

Let (x, y) be the coordinates of a point P w.r.t. the old axes and (x', y') be the coordinates w.r.t. the new axes.

The transformation formulas are:

x = x' cos(θ) - y' sin(θ)
y = x' sin(θ) + y' cos(θ)

These can be remembered using the following table:

Invariants in Orthogonal Transformation

Invariants are quantities in an equation that do not change their value after an orthogonal transformation (like rotation). This is a very useful concept for identifying conics.

Consider the general equation of the second degree: ax² + 2hxy + by² + 2gx + 2fy + c = 0.

After rotating the axes by an angle θ, the equation becomes a'x'² + 2h'x'y' + b'y'² + 2g'x' + 2f'y' + c' = 0.

The following quantities are **invariants**:

1. The degree of the equation: A second-degree equation remains second-degree.
2. The constant term (if origin is unchanged): c = c' (Note: The syllabus mentions orthogonal transformation, which is typically rotation. Translation *will* change the constant term).
3. The sum of the coefficients of x² and y²:

   a + b = a' + b'

4. The expression (ab - h²):

   ab - h² = a'b' - h'²

Pair of Straight Lines

A second-degree equation can sometimes represent two straight lines. We study two cases.

Homogeneous Equation of Second Degree

A homogeneous equation of the second degree is of the form:

ax² + 2hxy + by² = 0

This equation *always* represents a pair of straight lines passing through the **origin** (0, 0).

The two lines can be found by factoring the equation. If we divide by x², we get a quadratic in (y/x): b(y/x)² + 2h(y/x) + a = 0. If m₁ and m₂ are the roots (slopes) of this equation, the two lines are y = m₁x and y = m₂x.

Angle Between the Lines

The angle θ between the two lines ax² + 2hxy + by² = 0 is given by the formula:

tan(θ) = | 2 * sqrt(h² - ab) / (a + b) |

From this formula, we can determine the nature of the lines:

• Real and distinct lines: if h² > ab
• Real and coincident lines (one line): if h² = ab
• Imaginary lines (intersecting only at origin): if h² < ab

Special Cases:

• Perpendicular Lines (θ = 90°): The denominator must be zero.

  Condition: a + b = 0

  (i.e., Coefficient of x² + Coefficient of y² = 0)
• Parallel/Coincident Lines (θ = 0°): The numerator must be zero.

  Condition: h² - ab = 0 or h² = ab

General Equation of Second Degree

The full general equation is:

S = ax² + 2hxy + by² + 2gx + 2fy + c = 0

This equation represents a pair of straight lines (which may be parallel) if and only if a specific condition is met. This condition can be expressed in two ways:

Condition 1 (The "long" formula):

abc + 2fgh - af² - bg² - ch² = 0

Condition 2 (The determinant form):

The determinant Δ (delta) must be zero.

Δ =

a	h	g
h	b	f
g	f	c

Δ = a(bc - f²) - h(hc - fg) + g(hf - bg) = 0

Bisector of Angles Between Pair of Straight Lines

Given the pair of lines ax² + 2hxy + by² = 0, the combined equation of the two lines that bisect the angles between them is:

(x² - y²) / (a - b) = xy / h

If a = b, the equation becomes x² - y² = 0, or (x - y)(x + y) = 0, meaning the bisectors are y = x and y = -x.

If h = 0, the equation becomes xy = 0, meaning the bisectors are the axes x = 0 and y = 0.

Important Note: The two angle bisectors are always perpendicular to each other. We can check this using the condition A + B = 0. Rearranging the bisector equation: h(x² - y²) = (a - b)xy hx² - (a - b)xy - hy² = 0 Here, A = h and B = -h. A + B = h + (-h) = 0. Thus, they are always perpendicular.

Unit 1: Exam Quick Tips