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PAIR OF STRAIGHT LINES

Synopsis

      If l1x + m1y + n1 = 0 ........ (1) and l2x + m2y + n2 = 0 .......... (2) are equations of any two straight lines, then the combined equation of (1) & (2) is
               (l1x + m1y + n1)(l2x + m2y + n2) = 0.
The combined equation of (1) & (2) is called equation of pair of straight lines.

 

PAIR OF STRAIGHT LINES THROUGH THE ORIGIN
* Generally second degree homogeneous equation in 'x' and 'y' represents two straight lines through the origin.
* Let us consider the two straight lines through the origin as
     y = m1x ........ (1) and y = m2x ....... (2)
    Compare the (1) × (2) with ax2 + 2hxy + by2
then b[(y − m1x)(y − m2x)] = b 
y2 − (m1 + m2)xy + m1m2x2

* m1 + m2 =   and m1m2
*  Sum and product of slopes of two lines represented by the equation
      ax2 + 2hxy + by2 = 0 is   and    respectively.

 

ANGLE BETWEEN THE LINES REPRESENTED BY ax2 + 2hxy + by2 = 0
* If θ is the angle between the lines represented by the equation
     ax2 + 2hxy + by2 = 0, then


     
* If  two lines are perpendicular, then a + b = 0
* If  two lines are parallel (coincident), then h2 = ab


BISECTORS OF ANGLE BETWEEN LINES REPRESENTED BY
     ax2 + 2hxy + by2 = 0
* The pair of bisectors of angle between the lines represented by the equation
      ax2 + 2hxy + by2 = 0 is h(x2 − y2) = (a − b)xy

 

CONDITION FOR GENERAL SECOND DEGREE EQUATION IN 'x' AND 'y' to REPRESENT PAIR OF STRAIGHT LINES
* The second degree equation ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 represents two separate lines, then (i) h2 ab; g2 bc; f2 ac
(ii) ∆ = abc + 2fgh − af2 − bg2 − ch2 = 0
* Equation of pair of straight lines passing through origin and parallel to the pair of straight lines ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 is ax2 + 2hxy + by2 = 0
* If the lines l1x + m1y + n1 = 0 ................... (1)
                and l2x + m2y + n2 = 0 ................... (2)
      are the two separate lines of ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, then
               l1l2 = a; m1m2 = b; n1n2 = c
           l1m2 + l2m1 = 2h;   l1n2 + l2n1 = 2g;  m1n2 + m2n1 = 2f
* If 'θ' is the angle between the lines represented by the equation
    ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, then tan θ = 
* The point of intersection of two lines represented by the equation
      ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 is 

       
* If the equation S ≡ ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 represents a pair of parallel lines, then (i) h2 = ab
                   (ii) af2 = bg2
                   (iii) Distance between them 

                                
* The equations of the lines passing through (x0, y0) and parallel to the pair of lines
     ax2 + 2hxy + by2 = 0 is a(x − x0)2 + 2h (x − x0)(y − y0) + b(y − y0)2 = 0

* The equation of the lines passing through (x0, y0) and perpendicular to the lines ax2 + 2hxy + by2 = 0 is b(x − x0)2 − 2h(x − x0)(y − y0) + a(y − y0)2 = 0
* The equation of pair of straight lines joining the origin to the point of intersection of the curve
ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 ............... (1)
and lx + my + n = 0 is   ................ (2)     
ax2 + 2hxy + by2 + (2gx + 2fy) 

Posted Date : 19-02-2021

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