Question

The tangent to the circle c1 : x2+ y2− 2 x −1=0 at the point (2, 1) cuts off a chord of length 4 from a circle c2 whose centre is (3, −2). The radius of

Answer 1

Answer:

Radius of circle c₂ is √21/5 units

Step-by-step explanation:

Hi,

Tangent to the circle c₁: x² + y - 2x - 1 = 0 at the point (2, 1) will be

2x + (y + 1)/2 -(x + 2) - 1 = 0

⇒ x + y/2 - 5/2 = 0

⇒ 2x + y = 5 is the tangent to the circle at c₁.

Let line AB be segment of the chord 2x + y = 5 to the circle c₂.

Perpendicular from center will bisect the chord. Hence , if we drop a

perpendicular on AB say M be the foot of the perpendicular , then AM = MB

⇒ AM = MB = AB/2 = 4/2 = 2,

Perpendicular distance from center (3, -2) to the chord AB will be

|2(3) - 2 - 5|/√2² + 1² = 1/√5

and also if O is the center of the circle c₂, then

OA² = OM² + MA²

⇒ OA² = 1/5 + 4 = 21/5

⇒ OA = √(21/5).

Hence , the radius of circle c₂ is √21/5 units.

Hope, it helped !