Determining a Pair of Straight Lines from the Standard Equation of Conics: A Comprehensive Guide

Understanding Conic Sections and Their Representations

Conic sections, including circles, ellipses, parabolas, and hyperbolas, are represented by a general second-degree equation in two variables:

Ax2 Bxy Cy2 Dx Ey F 0

Deciding whether this equation represents a pair of straight lines involves a few key steps, notably, calculating the discriminant and factoring the equation.

Steps to Determine if Conics Represent a Pair of Straight Lines

1. Calculate the Discriminant

The discriminant, D, for a conic section is given by:

D B2 - 4AC

If the discriminant is zero, the conic is a pair of straight lines:

D 0

2. Factor the Equation

When D 0, the equation can be factored into two linear equations:

(y - m_1)(x - m_2) 0

Here, m_1 and m_2 are the slopes of the lines.

3. Using the Condition of Pair of Lines

By expanding and comparing coefficients with the original equation, you can find the values of m_1 and m_2.

Let's consider an example:

Example 1: x2 - 4xy y2 0

Here, A 1, B -4, and C 1

D (-4)2 - 4(1)(1) 16 - 4 12

Thus, D ≠ 0, and this conic does not represent a pair of straight lines.

Example 2: x2 4xy y2 0

Here, A 1, B 4, and C 1

D (4)2 - 4(1)(1) 16 - 4 12

Again, D ≠ 0, and this conic does not represent a pair of straight lines.

General Strategy to Resolve Into Linear Factors (Advanced)

To prove that a conic equation represents a pair of straight lines, we can use the following steps:

1. Shift the Origin

Shift the origin to a new point (H, K) such that the new coordinates (X, Y) are defined as:

x X - H, y Y - K

Substitute these into the original equation:

aX2 - 2hXY bY2 0

2. Solve for H and K

Set up the system of equations and solve for H and K:

aH - hK g 0

hH bK f 0

3. Verify the Constant Term

Ensure that the constant term vanishes:

aH2 - 2hHK bK2 - 2gH - 2fK c 0

This confirms that (H, K) lies on the pair of straight lines.

4. Rewrite the Equation

After solving for H and K, rewrite the equation in terms of X and Y:

aX2 - 2hXY bY2 0

Use the quadratic formula to rewrite this into linear factors:

(a_1X - b_1Y)(a_2X - b_2Y) 0

5. Transform Back to Original Coordinates

Transform the factors back to the original coordinates:

(a_1(x - H) - b_1(y - K))(a_2(x - H) - b_2(y - K)) 0

The individual straight lines are:

a_1x - b_1y - (a_1H - b_1K) 0

a_2x - b_2y - (a_2H - b_2K) 0

Conclusion

By following these steps, you can determine if a conic section represents a pair of straight lines and find the equations of the lines. The discriminant plays a crucial role in this process.