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EQUATION OF PAIR OF STRAIGHT LINES

3x + 2y – 1 = 0 and 4x – 3y – 7 = 0 2x – 3y + 5 = 0 and 4x – y – 3 = 0 x + y + 2 = 0 and x + 2y – 1 = 0 x + 3y = 0 and 3x + y = 0

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Equation of pair of straight lines(Second Degree Equation of Pair of Straight

Lines

IntroductionLet us consider two lines with equations3x + 2y – 1 = 0……………………………………………..(i) and 4x – 3y – 7 = 0…………………………………………..…(ii) Which are represented graphically as: (In graph)

Now, multiplying the equations (i) and (ii), we get (3x+2y – 1)(4x – 3y – 7) = 0

Or, 12x2 – 9xy – 21x + 8xy – 6y2 – 14y – 4x + 3y + 7 = 0Or, 12x2 – xy – 6y2 – 25x – 11y + 7 = 0…………….(iii)

Any point which satisfy the line (iii) also satisfy both equation (i) and (ii). Similarly, all the points which satisfy both line (I) and (ii) also satisfy line (iii)

In the adjoining graph, the point (1, -1) satisfy both the equation (i) and (ii), so the same point (1, -1) also satisfy the equation (iii).

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Conclusion: An equation obtained by multiplying two linear equations is called equation of a pair of straight lines. It is the combined form of the equations of two straight lines. It is also called general equation of second degree.

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Other examples :

Let us consider the following equations of straight lines

2x – 3y + 5 = 0……..(i) and 4x – y – 3 = 0…..(ii) Now, multiplying equations (i) and (ii), we get

(2x – 3y + 5 = 0)(4x – y – 3 = 0) = 08x2 – 14xy + 3y2 + 14x + 4y – 15 = 0………(iii)ax2 +2hxy + by2 + 2gx + 2fy + c = 0

Its graph is illustrated by geo -gebra

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General Equation of Second Degree

Let us consider two linear equationsa1 x + b1y + c1 = 0……..(i) and a2x + b2 y + c2 = 0……..(ii),

where a1, b1 , c1. a2, b2 , c2 R

Then their product will be(a1 x + b1y + c1 )(a2x + b2 y + c2 ) = 0Now, expanding, we geta1a2 x2 + (a1b2 + a2b1)xy + b1b2 y2 + (a1c2 + a2c1)x + (b1c2 + b2c1) y + c1c2 = 0

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Putting, a1a2 = a, (a1b2 + a2b1) = 2h , b1b2 = b , (a1c2 + a2c1) = 2g, (b1c2 + b2c1) = 2f , c1c2 = c

Then the equation becomes: ax2 + 2hxy + by2 + 2gx + 2fy + c = 0

a1a2 x2 + (a1b2 + a2b1)xy + b1b2 y2 + (a1c2 + a2c1)x + (b1c2 + b2c1) y + c1c2 = 0

Which is the requires general equation of second degree in x and y. It represents both equations (i) and (ii).

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Note: Every second degree equation in x and

y may not represent a pair of straight lines. If the left hand side of the equation (iii) can be resolved into two linear factors, it well represent only a pair of straight lines.

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Examples

x2 - 2xy + y2 - 3x + 3y = 0 represents a pair of straight lines. [i.e.(x – y) (x – y – 3) = 0]

6x2 + xy – 2y2 - 12x – y + 6 = 0 represent a pair of straight lines[i.e.(2x – y – 2)(3x + 2y – 3) =0]

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Equation may not represent a pair of lines

x2 - xy + y2 - 2x + 3y = 0 may not represent a pair of straight lines

x2 + xy – 2y2 - 2x – 4y + 3 = 0 may not represent a pair of straight lines.

Because: Left hand side of above equations have not the factors.

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Homogeneous equation of second degree in x and y

1. x2 – 7xy + 12y2 = 02. 4x2 + 5xy + y2 = 03. 33x2 – 44xy + 11y2 = 0These are in the form of ax2 +

2hxy + by2 = 0

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Homogeneous equation of second degree in x and y always represents a pair of straight line passes through the origin

2x2 + 3xy + y2 After factorize LHS:(x + y)(2x+y)= 0Either x + y = 0 or 2x + y = 0. These two line passes through the origin. (0,0) satisfies the equations

2x2 + 5xy +3y2 = 0. After factorize LHS(x + y)(2x +3 ) = 0Either x + y = 0 or 2x+ 3 = 0. These two lines also passes through the origin. (0,0) satisfies the equations.

That types of lines whose passes through the origin and represents a pair of line (two straight line) is called homogeneous equation of 2nd degree.