Question

If two of the lines represented by  

                      x4 + x3 y + cx2 y2 – xy3 + y4 = 0  

bisect the angle between the other two, then the value of c is 

Solution

Correct option is

– 6

Since the product of the slopes of the four lines represented by the given equation is 1 and a pair of lines represent the bisectors of the angles between the other two, the product of the slopes of each pair is –1. So let the equation of one pair be

ax2 + 2hxy – ay2 = 0.  

   

By hypothesis

            x4 + x3y + cx2y2 – xy3 + y4  

                 

                 

Compairing the respective co-efficients we get

           ah = 1 and c = –6ah = –6 

SIMILAR QUESTIONS

Q1

Equation of a line which is parallel to the line common to the pair of lines given by 6x2 – xy – 12y2 = 0 and 15x2 + 14xy – 8y2 = 0 and the sum of whose intercepts on the axes is 7, is 

Q2

If pairs of lines x2 + 2xy + ay2 = 0 and ax2 + 2xy + y2 = 0 have exactly one line in common then the joint equation of the other two lines is given by 

Q3

If the lines joining the origin to the intersection of the line y = mx + 2 and the curve x2 + y2 = 1 are at right angles, then 

Q4

Let PQR be a right angled isosceles triangle right angled at P (2, 1). If the equation of the line QR is 2x + y = 3, then the equation representing the pair of lines PQ and PR is  

Q5

If θ is an angle between the lines given by the equation 6x2 + 5xy – 4y2 + 7x + 13y – 3 = 0, then equation of the line passing through the point of intersection of these lines and making an angle θ with the positive x-axis is 

Q6

If one of the lines given by the equation 2x2 + axy + 3y2 = 0 coincide with one of those given by 2x2 + bxy – 3y2 = 0 and the other lines represented by them be perpendicular, then 

Q7

The equation x – y = 4 and x2 + 4xy + y2 = 0 represent the sides of  

Q8

If the equation of the pair of straight lines passing through the point (1, 1), one making an angle θ with the positive direction of x-axis and the other making the same angle with the positive direction of y-axis is x2 – (a + 2)xy + y2 + a(x + y – 1) = 0,  

a ≠ –2, then the value of sin 2θ is 

Q9

If θ1 and θ2 be the angles which the lines (x2 + y2) (cos2 θ sin2α + sin2 θ) = (x tan α – y sin θ)2 make with axis ofx, then if θ = π/6,

tan θ1 + tan θ2 is equal to  

Q10

The straight line is x + y = 0, 3x + y – 4 = 0 and x + 3y – 4 = 0 from a triangle which is