Question

C₁ and C₂ are circles of unit radius with centres at (0, 0) and (1, 0) respectively. C₃ is a circle of unit radius, passes through the centres of the circles C₁ and C₂ and have its centre above x-axis. Equation of the common tangent to C₁ and C₃ which does not pass through C₂ is –

• A. x – $\sqrt { 3 } y + 2 = 0$
• B. $\sqrt { 3 }$x – y + 2 = 0
• C. $\sqrt { 3 }$x – y – 2 = 0
• D. x + $\sqrt { 3 }$y + 2 = 0

Answer 1

Answer: (B)

$\sqrt { 3 }$x – y + 2 = 0

Answer 2

Equation of any circle through (0,0) and (1, 0) is

(x – 0) (x – 1) + (y – 0) (y – 0) + l = 0

Ž x² + y² – x + ly = 0

If it represents C₃, its radius = 1

Ž 1 = (1/4) + (l²/4) Ž l = ± $\sqrt { 3 }$

As the centre of C₃, lies above the x-axis, we take l = –$\sqrt { 3 }$and thus an equation of C₃ is x² + y² – x – $\sqrt { 3 }$y = 0

Since C₁ and C₃ intersect and are of unit radius, their common tangents are parallel to the line joining their centres (0, 0) and (1/2, $\sqrt { 3 }$/2).

So, let the equation of a common tangent be

$\sqrt { 3 }$x – y + k = 0 It will touch C₁, if $\left| \frac { \mathrm { k } } { \sqrt { 3 + 1 } } \right|$= 1 Ž k = ± 2

From the figure, we observe that the required tangent makes positive intercept on the y-axis and negative on the x-axis and hence its equation is – y + 2 = 0 .