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Question
For what values of k does the equation 12x2 + 2kxy + 2y2 +11x – 5y + 2 = 0 represent two straight lines
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Solution
The given equation of the pair of straight line is
12x2 + 2kxy + 2y2 + 11x – 5y + 2 = 0 .......(1)
Compare this equation with the equation
ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 .......(2)
a = 12, 2h = 2k, b = 2,
2g = 11, 2f = – 5, c = 2 ,
a =12, h = k, b = 2,
g = `11/2`, f = `- 5/2`, c = 2,
The condition for a second degree equation in x and y to represent a pair of straight lines is
abc + 2fgh – af2 – bg2 – ch2 = 0
`(12)(2)(2) + 2(- 5/2) (11/2)"k" - (12) (- 5/2)^2 - (2) (11/2)^2 - (2)("k")^2` = 0
`48 - (55"k")/2 - 12 xx 25/4 - 2 xx 121/4 - 2"k"^2` = 0
`48 - (55"k")/2 - 75 - 121/2 - 2"k"^2` = 0
96 – 55k – 150 – 121 – 4k2 = 0
– 4k2 – 55k – 175 = 0
4k2 + 55k + 175 = 0
k = `(- 55 +- sqrt(55^2 - 4(4)(175)))/(2(4))`
k = `(- 55 +- sqrt(3025 - 2800))/8`
k = `(- 55 +- sqrt(225))/8`
k = `(- 55 +- 15)/8`
k = `(- 55 + 15)/8` or k = `(- 55 - 15)/8`
k = `(- 40)/8` or k = `(- 70)/8`
k = – 5 or k = `(- 35)/4`
∴ The given equation represents a pair of straight lines when k = – 5 or k = `(- 35)/4`
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