Here, we need to find the single equation of the lines passing through origin and perpendicular to the lines represented by the equation
3x2 + xy – 10y2 = 0.
- The equation of a single line passing through the origin and perpendicular to the lines represented by the equation: 3x2 + xy – 10y2 = 0 is; 3y² - xy - 10x² = 0.
By factorisation; the lines represented by the equation 3x² + xy – 10y² = 0 can be gotten as;
- 3x² + 6xy - 5xy - 10y² = 0
- 3x² + 6xy - 5xy - 10y² = 03x(x + 2y) - 5y(x + 2y) = 0
- 3x² + 6xy - 5xy - 10y² = 03x(x + 2y) - 5y(x + 2y) = 03x -5y = 0 and x + 2y = 0
- 3x² + 6xy - 5xy - 10y² = 03x(x + 2y) - 5y(x + 2y) = 03x -5y = 0 and x + 2y = 0y = (3/5)x and y = -(1/2)x
- 3x² + 6xy - 5xy - 10y² = 03x(x + 2y) - 5y(x + 2y) = 03x -5y = 0 and x + 2y = 0y = (3/5)x and y = -(1/2)xm1 = 3/5 and m2 = (-1/2)
The product of the slope of 2 perpendicular lines is -1.
Therefore: the slope of the lines required are given by;
m1 = -1/(3/5) and m2 = -1/(-1/2)
Therefore, the slopes of the two lines are;
m1 and m2 are -5/3 and 2 respectively.
The equations of the lines since they pass through the origin, (0,0) is therefore;
y = (-5/3)x OR 3y + 5x = 0
y = 2x. OR y - 2x = 0
The equations of the two lines are 3y + 5x = 0. and y - 2x = 0.
Therefore; the single line is given as the product of both expressions; which yields;
3y² - xy - 10x² = 0.
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