For what values of $λ$ does the equation $6 x^{2} - x y + λ y^{2} = 0$ represent two perpendicular lines and two lines inclined at an angle of $\frac{π}{4}$ , respectively?

Answer (Detailed Solution Below)View answer ▾

Answer: Option C — -6 and -35

Detailed Solution

Compare with $a x^{2} + 2 h x y + b y^{2} = 0$ : $a = 6$ , $h = - \frac{1}{2}$ , $b = λ$ . Perpendicular lines need $a + b = 0$ , so $λ = - 6$ .

For angle $θ$ : $t an θ = \frac{2 h ^{2} - ab}{∣ a + b ∣}$ . With $θ = \frac{π}{4}$ : $2 \frac{1}{4} - 6 λ = ∣ 6 + λ ∣$ .

Squaring: $1 - 24 λ = 36 + 12 λ + λ^{2}$ , so $λ^{2} + 36 λ + 35 = 0$ , giving $λ = - 1$ or $λ = - 35$ . The matching option is $- 6$ and $- 35$ .

✓ Check

For

λ = - 35

h^{2} - ab = \frac{1}{4} + 210 = 210.25

210.25 = 14.5

, and

t an θ = \frac{29}{29} = 1

⚠ Common trap

(D)

λ = 1

gives

h^{2} - ab = \frac{1}{4} - 6 < 0

, so the equation has no real lines at all.

Quick tip

Perpendicular: $a + b = 0$ ; angle: $tan θ = \frac{2 h ^{2} - ab}{∣ a + b ∣}$

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Pair of Straight Lines — Preparation

This question tests your understanding of Pair of Straight Lines, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

A pair of straight lines can be represented by a single second-degree equation. This topic covers the homogeneous equation ax^2 + 2hxy + by^2 = 0, which represents two lines through the origin, and the general equation ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0. Exam questions test the condition for the general equation to represent a pair of lines, the angle between the lines (tan theta = 2 sqrt(h^2 - ab) / |a + b|), the conditions for the lines to be perpendicular (a + b = 0) or coincident (h^2 = ab), and the sum and product of the slopes (-2h/b and a/b). Knowing these results directly turns most questions into short calculations. Practise the pair of straight lines questions below, each with a step-by-step solution.

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For what values of $λ$ does the equation $6 x^{2} - x y + λ y^{2} = 0$ represent two perpendicular lines and two lines inclined at an angle of $\frac{π}{4}$ , respectively?

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For what values of $λ$ does the equation $6 x^{2} - x y + λ y^{2} = 0$ represent two perpendicular lines and two lines inclined at an angle of $\frac{π}{4}$ , respectively?