Question

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 

Solution

Correct option is

3x2 + 10xy + 3y2 = 0

Let y = mx be a line common to the given pairs of lines, then

am2 + 2m + 1 = 0 and m2 + 2m + a = 0  

   

    

         

But for a = 1, the two pairs have both the lines common, so

a = – 3 and the slope m of the line common to both the pairs is 1.  

Now x2 + 2xy + ay2 = x2 + 2xy – 3y2 = (x – y) (x + 3y)

and  ax2 + 2xy + y2 = –3x2 + 2xy + y2 = –(x – y) (3x + y)  

so the equation of the required lines is  

       (x + 3y) (3x + y) = 0 ⇒ 3x2 + 10xy + 3y2 = 0.    

SIMILAR QUESTIONS

Q1

If a line joining two points A (2, 0) and B (3, 1) is rotated about A in anticlockwise direction through and angle 15o, then equation of the line is the new position is

Q2

An equation of a line through the point (1, 2) whose distance from the point (3, 1) has the greatest value is

Q3

Let 0 < α < π/2 be a fixed angle. If P = (cos θ, sin θ) and Q = (cos (α – θ), sin (α – θ) then Q is obtained from P by

Q4

On the portion of the straight line x + y = 2 which is intercepted between the axes, a square is constructed, away from the origin, with this portion as one of its side. If p denotes the perpendicular distance of a side of this square from the origin, then the maximum value of p is 

Q5

The line x + y = 1 meets x-axis at A and y-axis at BP is the mid-point ofAB (Fig). P is the foot of the perpendicular from P to OAM1 is that from P1 to OPP2 is that from M1 to OAM2 is that from P2 to OPP3is that from M2 to OA and so on. If Pn denotes the nth foot of the perpendicular on OA from Mn – 1, then OPn =  

Q6

The line x + y = a, meets the axis of x and y at A and B respectively. A triangle AMN is inscribed in the triangle OABO being the origin, with right angle at NM and N lie respectively on OB and AB. If the area of the triangle AMN is 3/8 of the area of the triangle OAB, then AN/BN is equal to.

Q7

The point (4, 1) undergoes the following transformation successively.

(i) Reflection about the line y = x

(ii) Translation through a distance 2 units along the positive direction of x-axis.

(iii) Rotation through an anlge π/4 about the origin in the anticlockwise direction  

(iv) Reflection about x = 0

The final position of the given point is

 

Q8

A line cuts the x-axis at (7, 0) and the y-axis at B(0, –5). A variable linePQ is draw perpendicular to AB cutting the x-axis at P and the y-axis at in θ. If AQ and BP intersect at R, the locus of R is

Q9

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 

Q10

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