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Question
Find the value of k, if the following equation represents a pair of straight lines. Further, find whether these lines are parallel or intersecting, 12x2 + 7xy − 12y2 − x + 7y + k = 0
Solution
The equation of the given pair of straight lines is
12x2 + 7xy – 12y2 – x + 7y + k = 0 .......(1)
Compare this equation with the equation
ax2 + 2hxy + by2 + 2gx + 2f y + c = 0 .......(2)
a = 12, 2h = 7, b = – 12,
2g = – 1, 2f = 7, k = c
a = 12, h = `7/2`, b = – 12,
g = `- 1/2`, f = `7/2`, c = k
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
Substituting the values
`(12)(- 12)"k" + 2(7/2)(- 1/2)(7/2) - (12)(7/2)^2 - (- 12)(- 1/2)^2 - "k"(7/2)^2` = 0
`- 144"k" - 49/4 - 12 xx 49/4 + 12 xx 1/4 - "k" xx 49/4` = 0
`- 144"k" - 49/4 - 588/4 + 12/4 - (49"k")/4` = 0
`(- 576"k" - 49 - 588 + 12 - 49"k")/4` = 0
– 625k – 625 = 0
k = `- 625/625`
= – 1
Coefficient of x2 + coefficient of y2 = 12 – 12 = 0
∴ The given pair of straight lines are perpendicular and hence they are intersecting lines.
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