Question

The circles x² + y² + 2x − 2y + 1 = 0 and x² + y² − 2x − 2y + 1 = 0 touch each other : 

(a) externally at (0, 1) 

(b) internally at (0, 1) 

(c) externally at (1, 0) 

(d) internally at (1, 0)

Answer 1

Correct option (a) externally at (0, 1)

Explanation:

The equation of circle of the form

x² + y² + 2gx + 2fy + c = 0 has centre C (− g, − f) 

Therefore, the centre of the circle

x² + y² + 2x − 2y + 1 = 0 is C₁ (−1, 1)

x² + y² + 2x − 2y + 1 = 0 is C₁ (−1, 1) and centre of the circle x² + y² − 2x − 2y + 1 = 0 is C₂ (1, 1) and both have radii equal to 1. We have

Sum of radii = 1 + 1 = 2 

So, the two circles touch each other externally. 

The equation of the common tangent is obtained by subtracting the two equations. 

The equation of the common tangent is

Putting x = 0 in the equation of the either circle, we get

Hence, the points where the two circles touch is (0, 1).