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JEE Exam  >  JEE Questions  >  Let k1, k2 (k1 < k2) be two values of k fo... Start Learning for Free
Let k1, k2 (k1 < k2) be two values of k for which the expression x2 – y2 +kx +1 can be factorised into two real linear factors, then (k2 – k1) is equal to
  • a)
    2
  • b)
    – 2
  • c)
    0
  • d)
    4
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Let k1, k2 (k1 < k2) be two values of k for which the expression x2...
x2 + kx + 1 – y2 = 0

for real linear factors 4y2 + 0y + k2 – 4 must be a perfect square.
Hence D = 0 ⇒ 0 – 16(k2 – 4) = 0
∴ k = 2, – 2 ⇒ k1 = – 2 and k2 = 2
k2 – k1 = 2 – (–2) = 4  Ans.
Aliternatively: Comparing x2 – y2 + kx + 1, with Ax2 + 2Hxy + By2 + 2Gx + 2Fy + C;
we get A = 1, B = – 1, H = 0, G = k/2 , 
F = 0, C = 1
Now, using condition,
ABC + 2FGH – AF2 – BG2 – CH2 = 0, we get
⇒ k1 = – 2, k2 = 2
Hence, (k2 – k1) = 2 – (– 2) = 4. Ans.
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Most Upvoted Answer
Let k1, k2 (k1 < k2) be two values of k for which the expression x2...
Explanation:

Given Expression:
x^2 - y^2 + kx + 1

Factorization:
The given expression can be factorized into two real linear factors as:
(x + a)(x + b), where a and b are the roots of the quadratic equation x^2 - y^2 + kx + 1 = 0

Roots of the Quadratic Equation:
Using the quadratic formula, the roots can be calculated as:
x = (-k ± √(k^2 - 4))/2

Condition for Real Linear Factors:
For the expression to be factorized into real linear factors, the discriminant (k^2 - 4) should be greater than or equal to 0.

Calculation of k1 and k2:
Let k1 be the smallest value of k for which the expression can be factorized into real linear factors.
Let k2 be the larger value of k for which the expression can be factorized into real linear factors.

Calculation of k2 - k1:
(k2 - k1) = 4
Therefore, the correct answer is option 'D' (4).
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Let k1, k2 (k1 < k2) be two values of k for which the expression x2 – y2 +kx +1 can be factorised into two real linear factors, then (k2 – k1) is equal toa)2b)– 2c)0d)4Correct answer is option 'D'. Can you explain this answer?
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