Question: Find the equation of straight lines passing through the origin and perpendicular to the given line: $5x² -7xy -3y² = 0$.

Solution:
Given,

Equation of a single line is $5x² -7xy -3y²= 0$ - (i)

Comparing equation (i) with ax²+ 2hxy +by²=0, we get,
$a = 5, 2hxy = -7xy, and b = -3$

We know,
The equation of any line passing through the origin and perpendicular to the line represented by ax² +2hxy +by² = 0 is bx² -2hxy +ay²= 0.

Put value of a, 2hxy and b in bx² -2hxy +ay² = 0, we get,
$or, -3x² - (-7xy) + 5y² = 0$
$or, -3x² +7xy + 5y² = 0$ 
$or, 3x² -7xy -5y² = 0$ is the required equation.

Therefore, the required single equation of straight lines is 3x² -7xy -5y² = 0.

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