Question 1 of 9 · Pair of Straight Lines

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If two lines represented by $x^{4} + x^{3} y + c x^{2} y^{2} - x y^{3} + y^{4} = 0$ bisect the angle between the other two, then the value of c is

Answer (Detailed Solution Below)View answer ▾

Answer: Option D — -6

Detailed Solution

The two bisecting lines are perpendicular, so they form $x^{2} + k x y - y^{2} = 0$ . Write the quartic as $(x^{2} + k x y - y^{2}) (a x^{2} + p x y + b y^{2})$ ; matching $x^{4}$ and $y^{4}$ gives $a = 1$ , $b = - 1$ .

Expanding $(x^{2} + k x y - y^{2}) (x^{2} + p x y - y^{2})$ : the $x^{3} y$ coefficient is $k + p = 1$ and the $x^{2} y^{2}$ coefficient is $k p - 2 = c$ .

For the pair $x^{2} + p x y - y^{2} = 0$ the angle bisectors are $\frac{x ^{2} - y ^{2}}{2} = \frac{x y}{p /2}$ , i.e. $x^{2} - \frac{4}{p} x y - y^{2} = 0$ . Matching with $x^{2} + k x y - y^{2}$ gives $k p = - 4$ , so $c = - 4 - 2 = - 6$ .

✓ Check

k + p = 1

and

k p = - 4

give

k = \frac{1 + 17}{2}

and

p = \frac{1 - 17}{2}

, which are real, and then

(x^{2} + k x y - y^{2}) (x^{2} + p x y - y^{2})

has

x^{2} y^{2}

coefficient

- 4 - 2 = - 6

⚠ Common trap

The

x^{2} y^{2}

coefficient also receives

- 1

from each of the two end products, so

c = k p - 2

, not

k p

; forgetting the

- 2

gives

c = - 4

, which is not an option.

Quick tip

$(x^{2} + k x y - y^{2}) (x^{2} + p x y - y^{2})$ : $k + p = 1$ , $k p = - 4$ , $c = k p - 2$

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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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If two lines represented by $x^{4} + x^{3} y + c x^{2} y^{2} - x y^{3} + y^{4} = 0$ bisect the angle between the other two, then the value of c is

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If two lines represented by $x^{4} + x^{3} y + c x^{2} y^{2} - x y^{3} + y^{4} = 0$ bisect the angle between the other two, then the value of c is