Question

If d denotes the distance of origin from the common tangent to circles S1 & S2 where S1 : x2 + y2 – 2x – 4y = 44 & S2 : x2 + y2 – 8x – 12y + 48 = 0, then d is equal to

Answer 1

the distance between origin and the common tangent is 9.2

Given - equation of two circles S1: x² + y² -2x -4y = 44

S2 : x² + y² -8x -12y + 48 = 0

To Find - distance of origin from common tangent

Solution -

radius of circle1 = √ (g²+f² -c)  where g = -1 , f = -2 c = -44

radius of circle1 = 7

center of circle 1 = (-g , -f) = ( 1 ,2)

similarly in circle2  g = -4 , f = -6 , c =48

radius of circle 2 = 2

center of circle 2 = ( 4, 6)

now,

distance between centers of circle = ∣r1 -r2∣

so equation of common tangent is

S1 - S2 = 0

6x +8y =92

now we will calculate the distance of this tangent from origin using formula

d = ∣Ax1 +By1 +C∣/ √A² +B²

putting the values

A = 6 , B = 8 , C= -92  and (x1,y1) =(0,0)

d = 92/10

d = 9.2

the distance between origin and the common tangent is 9.2

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