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).

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Question

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).

Solution

Correct option is

2x2 + 2y2 – x – 7y = 0

 

Equations of circles whose limiting points are (1, 2) and (4, 3) are

                     (x – 1)2 + (y – 2)2 = 0 

or               x2 + y2 – 2x – 4y + 5 = 0                      …(1)

and               (x – 4)2 + (y – 3)2 = 0

or              x2 + y2 – 8x – 6y + 25 = 0                     …(2)

Therefore the corresponding system of co-axial circle is

    (x2 + y2 – 2x – 4y + 5) + λ(x2 + y2 – 8x – 6y + 25) = 0       …(3)

It passes through origin then

                      5 + 25λ = 0

  

Substituting the value of λ in (3), the required circle is

        5(x2 + y2 – 2x – 4y + 5) – (x2 + y2 – 8x – 6y + 25) = 0

or     4x2 + 4y2 – 2x – 14y = 0

or     2x2 + 2y2 – x – 7y = 0

Testing

SIMILAR QUESTIONS

Q1

Find the circle whose diameter is the common chord of the circles x2 + y2+ 2x + 3y + 1 = 0 and x2 + y2 + 4x + 3y + 2 = 0.

Q2

 

Find the angle between the circles

Q3

Find the equation of the circle which cuts the circle x2 + y2 + 5x + 7y – 4 = 0 orthogonally, has its centre on the line x = 2 and passes through the point (4, –1).

Q4

 

Find the equations of the two circles which intersect the circles

        x2 + y2 – 6y + 1 = 0 and x2 + y2 – 4y + 1 = 0  

Orthogonally and touch the line 3x + 4y + 5 = 0.   

Q5

 

Find the radical centre of circles x2 + y2 + 3x + 2y + 1 = 0,

x2 + y2 – x + 6y + 5 = 0 and x2 + y2 + 5x – 8y + 15 = 0. Also find the equation of the circle cutting them orthogonally.

Q6

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.

Q7

 

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      

Q8

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.

Q9

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

Q10

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