Q. 15

# Find the equation of a circle passing through the origin and intercepting lengths a and b on the axes.

Answer :

From the figure

AD = b units and AE = a units.

D(0, b), E(a, 0) and A(0, 0) lies on the circle. C is the centre.

The general equation of a circle: (x - h)^{2} + (y - k)^{2} = r^{2}

…(i), where (h, k) is the centre and r is the radius.

Putting A(0, 0) in (i)

(0 - h)^{2} + (0 - k)^{2} = r^{2}

h^{2} + k^{2} = r^{2} …(ii)

Similarly putting D(0, b) in (i)

(0 - h)^{2} + (b - k)^{2} = r^{2}

h^{2} + k^{2} + b^{2} - 2kb = r^{2}

r^{2} + b^{2} - 2kb = r^{2}

b^{2} - 2kb = 0

(b- 2k)b = 0

Either b = 0ork =

Similarly putting E(a, 0) in (i)

(a - h)^{2} + (0 - k)^{2} = r^{2}

h^{2} + k^{2} + a^{2} - 2ha = r^{2}

r^{2} + a^{2} - 2ha = r^{2}

a^{2} - 2ha = 0

(a- 2h)a = 0

Either a = 0orh =

Centre = C

r^{2} = h^{2} + k^{2}

Putting the value of r^{2} , h and k in equation (i)

(x - h)^{2} + (y - k)^{2} = r^{2}

which is the required equation.

Rate this question :

Show that the quadrilateral formed by the straight lines x – y = 0, 3x + 2y = 5, x – y = 10 and 2x + 3y = 0 is cyclic and hence find the equation of the circle.

RS Aggarwal - MathematicsIf ( - 1, 3) and (∝, β) are the extremities of the diameter of the circle x^{2} + y^{2} – 6x + 5y – 7 = 0, find the coordinates (∝, β).

Find the equation of the circle which passes through the points A(1, 1) and B(2, 2) and whose radius is 1. Show that there are two such circles.

RS Aggarwal - MathematicsFind the equation of the circle concentric with the circle x^{2} + y^{2} – 6x + 12y + 15 = 0 and of double its area.