Q. 15

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

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 = r2

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

Putting A(0, 0) in (i)

(0 - h)2 + (0 - k)2 = r2 h2 + k2 = r2 …(ii)

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

(0 - h)2 + (b - k)2 = r2 h2 + k2 + b2 - 2kb = r2 r2 + b2 - 2kb = r2 b2 - 2kb = 0 (b- 2k)b = 0

Either b = 0ork = Similarly putting E(a, 0) in (i)

(a - h)2 + (0 - k)2 = r2 h2 + k2 + a2 - 2ha = r2 r2 + a2 - 2ha = r2 a2 - 2ha = 0 (a- 2h)a = 0

Either a = 0orh = Centre = C r2 = h2 + k2 Putting the value of r2 , h and k in equation (i)

(x - h)2 + (y - k)2 = r2   which is the required equation.

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