Question

Let ABCD be a square of side of unit length. Let a circle C₁ centered at A with unit radius is drawn. Another circle C₂ which touches C₁ and the lines AD and AB are tangent to it, is also drawn. Let a tangent line from the point C to the circle C₂ meet the side AB at E. If the length of EB is `α + sqrt(3)β`, where α, β are integers, then α + β is equal to ______.

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Answer 1

Let ABCD be a square of side of unit length. Let a circle C₁ centered at A with unit radius is drawn. Another circle C₂ which touches C₁ and the lines AD and AB are tangent to it, is also drawn. Let a tangent line from the point C to the circle C₂ meet the side AB at E. If the length of EB is `α + sqrt(3)β`, where α, β are integers, then α + β is equal to 1.

Explanation:

C₁: Center = A(0, 0)

Radius = 1 unit = AD

OD = AM = MO = r

⇒ AO = `sqrt(2)r`

∵ AD = 1

AO + OD = 1

⇒ `sqrt(2) + r` = 1

⇒ r = `1/(sqrt(2) + 1)`

⇒ r = `sqrt(2) - 1`

So for circle C₂: Center is (r, r) and Radius is `sqrt(2) - 1`.

So, equation circle will be `(x - r)^2 + (y - r)^2 = (sqrt(2) - 1)^2`

Let slope of line, which passes through point C(1, 1) and Eis m. Then equation of line will be y – 1 = m(x – 1);

m > 0 (From figure) 

mx – y + 1 – m = 0   ...(i)

Line CE will be tangent to circle C₂ if OQ = r; where OQ is perpendicular distance of point O(r, r) from the line CE.

Hence, `|(mr - r + 1 - m)/sqrt(m^2 + 1)| = sqrt(2) - 1; r = sqrt(2) - 1`

⇒ `|((r - 1)(m - 1))/sqrt(m^2 + 1)| = sqrt(2) - 1`

As r = `sqrt(2) - 1`

⇒ `|((sqrt(2) - 2)(m - 1))/sqrt(m^2 + 1)| = sqrt(2) - 1`

⇒ `|(sqrt(2)(m - 1))/sqrt(m^2 + 1)|` = 1

Squaring both the sides

⇒ `(2(m - 1)^2)/(m^2 + 1)` = 1

⇒ m² – 4m + 1 = 0

⇒ m = `2 +- sqrt(3)`

 As m > 0, `2 - sqrt(3)` rejected

m = `2 + sqrt(3)`

Now using equation (i)

`(2 + sqrt(3))x - y + 1 - (2 + sqrt(3))` = 0

For E, y = 0

x = `(sqrt(3) + 1)/(2 + sqrt(3))`

⇒  x = `sqrt(3) - 1`

`E(sqrt(3) - 1, 0)` and B(1, 0)

EB = OB – OE

EB = `1 - (sqrt(3) - 1)`

EB = `2 - sqrt(3)`

`α + sqrt(3)β = 2 + sqrt(3)(-1)`; EB = ` α + sqrt(3)β`

After comparing we get

α = 2, β = –1

α + β = 1.