Question

prove that the length of the common chord of the two circles whose equations are (x-a)²-(y-b)²=c² and (x-b)²-(y-d)²=c² is ✓(4c²-2(a-b)²)​

Answer 1

Answer:

R.E.F image

S

1

=(x−a)

2

+(y−b)

2

=c

2

S

2

=(x−b)

2

+(y−a)

2

=c

2

∴Eq

n

of common chord

→S

1

−S

2

=0

⇒(x−a)

2

+(y−b)

2

−(x−b)

2

−(y−a)

2

=0

⇒−2xa−2yb+2xb+2ya=0

⇒(a−b)(y−x)=0

eq

n

of common chord ⇒y=x

C

1

M= length of ⊥ from C

1

(a,b) an line PQx−y=0

Length C

1

M=

2

∣a−b∣

C

1

P= radius of 1

st

circle = C

∴ In ΔPC

1

M,PM=

(PC

1

)

2

+(C

1

M)

2

⇒

c

2

−

2

(a−b)

2

PQ=2PM=2

C

2

−

2

(a−b)

2

PQ=

4C

2

−2(a−b)

2

(option