Question

Find the equation of the circle circumscribing the quadrilateral formed by the lines in order are 5x + 3y – 9 = 0, x – 3y = 0, 2x – y = 0, x + 4y – 2 = 0 without finding the vertices of quadrilateral.

Solution

Correct option is

 

Let the given lines are represented by L1L2L3 and L4, then

                     

                     

                     

                     

Equation of second degree conic circumscribing a quadrilateral whose sides are L1 = 0, L2 = 0, L3 = 0 and L4 = 0

is                 

For circle, coefficient of x2 = coefficient of y2    

   

and coefficient of xy = 0   

∴                    1 + λ = 0 which is true from (2).

Substituting the value of λ in (1), the required circle is  

              

SIMILAR QUESTIONS

Q1

 

Find the co-ordinates of the limiting points of the system of circles determined by the two circles

  x2 + y2 + 5x + y + 4 = 0 and x2 + y2 + 10x – 4y – 1 = 0      

Q2

If the origin be one limiting point of a system of co-axial circles of whichx2 + y2 + 3x + 4y + 25 = 0 is a member, find the other limiting point.

Q3

Find the radical axis of co-axial system of circles whose limiting points are (–1, 2) and (2, 3).

Q4

Find the equation of the circle which passes through the origin and belongs to the co-axial of circles whose limiting points are (1, 2) and (4, 3).

Q5

Find the equation of the image of the circle x2 + y2 + 16x – 24y + 183 = 0 by the line mirror 4x + 7y + 13 = 0.

Q6

Find the area of the triangle formed by the tangents drawn from the point (4, 6) to the circle x2 + y2 = 25 and their chord of contact. Also find the length of chord of contact.

Q7

 

Find the lengths of external and internal common tangents to two circlesx2 + y2 + 14x – 4y + 28 = 0 and x2 + y2 – 14x + 4y – 28 = 0.      

 

Q8

Find the lengths of common tangents of the circles x2 + y2 = 6x and x2 +y2 + 2x = 0.  

Q9

 

Find the equation of the circle circumscribing the triangle formed by the lines:

         x + y = 6, 2x + y = 4 and x + 2y = 5,

Without finding the vertices of the triangle.

Q10

Find the equation of a circle which touches the x-axis and the line 4x – 3y+ 4 = 0. Its centre lies in the third quadrant and lies on the line x – y – 1 = 0.