Find the combined equation of the bisectors of the angles between the lines represented by 5x²+6xy–y²=0

The given equation of the pair of straight lines is:

5x² + 6xy - y² = 0

Step 1: Finding the Individual Lines
The general equation of a pair of straight lines is:

Ax² + 2Hxy + By² = 0

Comparing with the given equation:

A = 5, H = 3, B = -1

The equation of the bisectors of the angles between two lines given by Ax² + 2Hxy + By² = 0 is:

(x² - y²) / (A - B) = xy / H

Substituting the values of A, B, and H:

(x² - y²) / (5 - (-1)) = xy / 3

(x² - y²) / 6 = xy / 3

Step 2: Cross Multiplication
Cross multiplying:

3(x² - y²) = 6xy

Expanding:

3x² - 3y² = 6xy

Rearranging:

3x² - 6xy - 3y² = 0

Dividing by 3:

x² - 2xy - y² = 0

Final Answer
The combined equation of the bisectors of the angles between the given lines is:

x² - 2xy - y² = 0