Question

Find the equation of the circle passing through (1, 0) and (0, 1) and having the smallest possible radius.

Solution

Correct option is

x2 + y2 – x – y = 0

 

Let the equation of circle be

                    x2 + y2 + 2gx + 2fy + c = 0           …(1)

since circle (1) passes through (1, 0) and (0, 1) then

                     

                                               

For minimum radius, c must be equal to zero, then from (2) and (3), 

                      

Equation of required circle, from (1), is 

                  x2 + y2 – x – y = 0

SIMILAR QUESTIONS

Q1

 

Let a circle be given by

                   2x (x – a) + y(2y – b) = 0            (a ≠ 0, b ≠ 0)

Find the condition on a and b if two chords, each bisected by the x-axis, can be drawn to the circle from (ab/2).

Q2

The centre of the circle S = 0 lie on the line 2x – 2y + 9 = 0 and S = 0 cuts orthogonally the circle x2 + y2 = 4. Show that circle S = 0 passes through two fixed points and find their co-ordinates.

Q3

 be a given circle. Find the locus of the foot of perpendicular drawn from origin upon any chord of Swhich subtends a right angle at the origin.

Q4

 be a given circle. Find the locus of the foot of perpendicular drawn from origin upon any chord of Swhich subtends a right angle at the origin.

Q5

P is a variable on the line y = 4. Tangents are drawn to the circle x2 + y2= 4 from P to touch it at A and B. The perpendicular PAQB is completed. Find the equation of the locus of Q.

Q6

 

Find the condition on abc such that two chords of the circle

                x2 + y2 – 2ax – 2by + a2 + b2 – c2 = 0  

passing through the point (ab + c) are bisected by the line y = x.  

Q7

 

Find the limiting points of the circles 

       (x2 + y2 + 2gx + c) + λ(x2 + y2 + 2fy + d) = 0

Q8

The circle x2 + y2 – 4x – 8y + 16 = 0 rolls up the tangent to it at  by 2 units, assuming the x-axis as horizontal, find the equation of the circle in the new position.  

Q9

 

Find the equation of the circle of minimum radius which contains the three circles 

                   x2 – y2 – 4y – 5 = 0 

               x2 + y2 + 12x + 4y + 31 = 0  

and         x2 + y2 + 6x + 12y + 36 = 0

Q10

 

If the line x cos α + y sin α = p cuts the circle x2 + y2 = a2 in M and N, then show that the circle, whose diameter is MN, is