Question

If the slopes of the lines given by 3x² + kxy - y² = 0 differ by 4, then the value of k is:
1. k = ±1
2. k = ​±3
3. k = ±2
4. k = ±4

Answer 1

Correct Answer - Option 3 : k = ±2

Concept:

• For a pair of lines ax2 + 2hxy + by2 = 0:
  • Sum of the slopes is: \(\rm -\frac{2h}{b}\).
  • Product of the slopes is: \(\rm \frac{a}{b}\).

 

Calculation:

Comparing the equation 3x2 + kxy - y2 = 0 with the general equation of a pair of lines ax2 + 2hxy + by2 = 0, we can say that: a = 3, b = -1 and 2h = k.

Let's say that the slopes of the pair of lines are m and n.

Product of the slopes:

\(\rm mn=\frac{a}{b}=\frac3{-1}\)

⇒ mn = -3                ...(1)

It is given that |m - n| = 4.

Squaring both sides, we get:

⇒ m² + n² - 2mn = 16

⇒ m2 + n2 - 2(-3) = 16                ...[Using equation (1)]

⇒ m2 + n2 = 10

Adding 2mn to both sides, we get:

⇒ m2 + n2 + 2mn = 10 + 2(-3)                ...[Using equation (1)]

⇒ (m + n)2 = 4

⇒ m + n = ±2                ...(2)

Now, sum of the slopes:

\(\rm m+n=-\frac{2h}{b}=-\frac{k}{-1}\)

⇒ m + n = k

Using equation (2), we get:

⇒ k = ±2.

 

The angle θ between the pair of lines ax² + 2hxy + by² = 0, is given by: \(\rm \tan \theta=\frac{2\sqrt{h^2-ab}}{a+b}\).

• The product of the slopes of a pair of perpendicular lines is -1.
• The slopes of parallel lines are equal.