Question

The circles which can be drawn, to pass through (1, 0) and

(3, 0) and to touch the y-axis, intersect at an angle q, then cos q is equal to-

• A. $\frac { 1 } { 2 }$
• B. – $\frac { 1 } { 2 }$
• C. $\frac { 1 } { 4 }$
• D. – $\frac { 1 } { 4 }$

Answer 1

Answer: (B)

– $\frac { 1 } { 2 }$

Answer 2

Let A Ί (1, 0), B Ί (3, 0) and C₁, C₂ be the centre of circles passing through A, B and touching the y-axis at P₁ and P₂. If r be the radius of circle (note that radius of both circles will be same), C₁A = C₂A = r = OD = 2 and

C₁ Ί (2, h), where h² = OC₁² – AD² = 4 – 1 = 3.

ή C₁ Ί (2,$\sqrt { 3 }$), C₂ Ί (2, – $\sqrt { 3 }$)

If Π C₁AC₂ = a

ή cos a = $\frac { 1 } { 2 }$