Question

Find the equation of the circle which touches the axes and whose centre lies on the line x – 2y = 3.

Solution

Correct option is

x2 + y2 – 2x + 2y + 1 = 0

 

Since the circle touches both the axes, let the radius of the circle by a, then

Case I: If centre (aa) but given centre lies on

                 x – 2y = 3

∴              a – 2a = 3

∴                      a = –3

∴              Centre = (–3, –3)

and           radius = |–3| = 3  

∴ Equation or circle is  

                 (x + 3)2 + (y + 3)2 = 32

and           x2 + y2 + 6x + 6y + 9 = 0

Case II: If centre (–aa) but centre lies on x – 2y = 3

Then          Centre = (1, –1) and radius = |–1| = 1

∴ Equation of circle is       (x – 1)2 + (y + 1)2 = 1

or              x2 + y2 – 2x + 2y + 1 = 0

Case III: If the centre = (–a, – a)

But centre lies on x – 2y = 3

∴              –a + 2a = 3 

∴                        a = 3  

Then centre (–3, –3) and radius = |3| = 3

∴ Equation of circle is

                 (x + 3)2 + (y + 3)2 = 32

or              x2 + y2 + 6x + 6y + 9 = 0

Case IV: If centre = (a, –a) but centre lies on x – 2y = 3

or             a + 2a = 3

∴                      a = 1

Then centre = (1, –1) and radius = 1

∴ Equation of circle is         (x – 1)2 + (y + 1)2 = 1

or              x2 + y2 – 2x + 2y + 1 = 0

SIMILAR QUESTIONS

Q1

The sides of a square are x = 2, x = 3, y = 1and y = 2. Find the equation of the circle drawn on the diagonals of the square as its diameter.

Q2

Find the equation of the circum circle of the quadrilateral formed by the four lines ax + by ± c = 0 and bx – ay ± c = 0.

Q3

The abscissa of two points A and B are the roots of the equation x2 + 2ax – b2 = 0 and their ordinates are the roots of the equation x2 + 2px –q2 = 0. Find the equation and the radius of the circle with AB as diameter.

Q4

Find the equation of the circle which passes through the points (4, 1), (6, 5) and has its centre on the line 4x + y = 16.

Q5

Find the equation of the circle passing through the three non-collinear points (1, 1), (2, –1) and (3, 2).

Q6

Show that the four points (1, 0), (2, –7), (8, 1) and (9, –6) are concyclic.

Q7

Find the equation of the circle whose diameter is the line joining the points (–4, 3) and (12, –1). Find also the intercept made by it on y-axis.

Q8

Find the equation of the circle which touches the axis of y at a distance of 4 units from the origin and cuts the intercept of 6 units from the axis of x.

Q9

Find the equation of the circle which passes through the origin and makes intercepts of length a and b on the x and y axes respectively.

Q10

A circle of radius 2 lies in the first quadrant and touches both the axes of co-ordinates. Find the equation of the circle with centre at (6, 5) and touching the above circle externally.