EQUATION OF PAIR OF STRAIGHT LINES PERPENDICULAR TO TWO GIVEN LINES

Question :

Find the equation of the pair of straight lines passing through the point (1, 3) and perpendicular to the lines 2x - 3y + 1 = 0 and 5x + y - 3 = 0

Answer :

First let us first the separate equations of pair of straight lines, and find the product to get the required equation.

Separate equation 1 of the pair of straight lines which is perpendicular to 2x - 3y + 1  =  0

3x + 2y + a  =  0

The point (1, 3) lies on the line 3x + 2y + a  =  0.

3(1) + 2(3) + a  =  0

3 + 6 + a = 0

a  =  -9

3x + 2y - 9  =  0  -----(1)

Separate equation 2 of the pair of straight lines which is perpendicular to 5x + y - 3  =  0

x - 5y + b  =  0

The point (1, 3) lies on the line x - 5y + b  =  0.

1 - 5(3) + b  =  0

1 - 15 + a = 0

a  =  14

x - 5y + 14  =  0 -----(2)

Product of (1) and (2)

(3x + 2y - 9)(x - 5y + 14)  =  0

3x² - 15xy + 42x + 2xy - 10y² + 28y - 9x + 45y - 126  =  0

3x² - 13xy - 10y² + 33x + 72y - 126  =  0

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