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.

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Question

 

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.

Solution

Correct option is

 

Let the given lines represented by L1L2 and L3 then  

       L­1 ≡ x + y – 6 = 0

       L2 ≡ 2x + y – 4 = 0

       L3 ≡ x + 2y – 5 = 0   

Equation of conic of second degree, whose sides are L1 = 0, L2 = 0 and L3 = 0, is 

              

             

For circle, coefficient of x2 = coefficient of y2   

  

 

And coefficient of xy = 0  

  

     

Substituting the values of λ and μ from (3) and (4) in (2), then

  

Testing

SIMILAR QUESTIONS

Q1

Find the radical centre of three circles described on the three sides 4x – 7y + 10 = 0, x + y – 5 = 0 and 7x + 4y – 15 = 0 of a triangle as diameters.

Q2

 

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      

Q3

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.

Q4

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

Q5

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).

Q6

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

Q7

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.

Q8

 

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

 

Q9

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

Q10

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.