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If the equation

3x2 + 7xy + 2y2 + 5x + 5y + k = 0

represents a pair of straight lines, then the value of k is


1. 1
2. 2
3. 3
4. 4

1 Answer

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Best answer
Correct Answer - Option 2 : 2

Concept:

Let second-degree equation be,

ax+ by+ 2hxy + 2gx + 2fy + c = 0

Discriminant is calculated as:

Δ = \(\left| {\begin{array}{*{20}{c}} a&h&{\rm{g}}\\ h&{\rm{b}}&{\rm{f}}\\ {\rm{g}}&{\rm{f}}&{\rm{c}} \end{array}} \right| \)

Δ = abc + 2fgh - af- bg- ch2

Case: 1

If discriminant (Δ) of this equation doesn’t equal to zero (Δ ≠ 0)

  • If h2 – ab > 0, it represents a hyperbola and a rectangular hyperbola (a + b = 0).
  • If h2 – ab = 0, it represents a parabola.
  • If h2 – ab < 0, it represents an ellipse. (a ≠ b)
  • If h2 – ab < 0, it represents a circle. (a = b)


Case: 2

If discriminant (Δ) of this equation equal to zero (Δ = 0)

  • If h2 – ab > 0, it represents two distinct real lines or pair of perpendicular straight lines
  • If h2 – ab = 0, it represents a parallel lines.
  • If h2 – ab < 0, it represents non-real lines.

 

Calculation:

Given: Second degree equation, 3x2 + 7xy + 2y2 + 5x + 5y + k = 0

Compare with second-degree equation ax+ 2hxy + by2 + 2gx + 2fy + c = 0

So, a = 3, h = 3.5, b = 2, g = 2.5, f = 2.5, c = k

The given equation represents the pair of straight lines so;

Δ = 0

abc + 2fgh - af- bg- ch2 = 0

3 x 2 x k + 2 x 2.5 x 2.5 x 3.5 - 3 x (2.5)2 - 2 x (2.5)2 - k x (3.5)2 = 0

6k + 43.75 - 18.75 - 12.5 - 12.25k = 0

6.25k = 12.5

k = 2

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