Question

If 3x^2 +10xy +3y^2 -5x-21y+k=0 represents a pair of straight lines then find value of k

Answer 1

Pair of lines L₁L₂:  3 x² + 10 x y + 3 y² - 5 x - 21 y + k = 0     ---- (1)
3 x² + 10 x y + 3 y² = (3x + y) (x + 3 y)
=>   L₁L₂ = (3x + y + p) (x + 3y + q)

Expand:   3x² + 10xy + 3y² + (p+3q) x + (3p+q) y + p q = 0   ---- (2)
 
Compare (1) and (2):  
         p + 3q = - 5         --- (3)
         3 p + q = -21      --- (4)
         pq = k               ---- (5)

Solving (3) and (4) we get:   -8 p = 58,   p = -29/4
    So    q = -21 - 3 p = - 3(7+p) = 3/4

Using (5),    k = p q = - 87/16.
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Alternatively,  we can use a standard formula to solve it.

If   a x² + 2h x y + b y² + 2 g x + 2fy + c = 0  represents a pair of straight lines, then
    abc + 2 fgh - af² - b g² - ch² = 0.  
    h² ≥ ab.         g² ≥ ac     f² ≥ b c
 
Given : a = 3.  b = 3.  h = 10/2=5.  g = -5/2.   f = -21/2.   c = k.

=>   3*3*k + 2 *21/2*5/2 * 5 - 3* 21²/4 - 3 * 5²/4 - k * 5² = 0
        16 k = 525/2 - 1323/4 - 75/4

=>       k = -87/16

Answer 2

Pair of lines L₁L₂:  3 x² + 10 x y + 3 y² - 5 x - 21 y + k = 0     ---- (1)

3 x² + 10 x y + 3 y² = (3x + y) (x + 3 y)

=>   L₁L₂ = (3x + y + p) (x + 3y + q)

Expand:   3x² + 10xy + 3y² + (p+3q) x + (3p+q) y + p q = 0   ---- (2)

 

Compare (1) and (2):  

         p + 3q = - 5         --- (3)

         3 p + q = -21      --- (4)

         pq = k               ---- (5)

Solving (3) and (4) we get:   -8 p = 58,   p = -29/4

    So    q = -21 - 3 p = - 3(7+p) = 3/4

Using (5),    k = p q = - 87/16.

============================

Alternatively,  we can use a standard formula to solve it.

If   a x² + 2h x y + b y² + 2 g x + 2fy + c = 0  represents a pair of straight lines, then

    abc + 2 fgh - af² - b g² - ch² = 0.  

    h² ≥ ab.         g² ≥ ac     f² ≥ b c

 

Given :

• a = 3. 
• b = 3.
• h = 10/2=5. 
• g = -5/2. 
•  f = -21/2.   
• c = k.

=>   3*3*k + 2 *21/2*5/2 * 5 - 3* 21²/4 - 3 * 5²/4 - k * 5² = 0

=> 16 k = 525/2 - 1323/4 - 75/4

=>  k = -87/16