Let f (x + y) = f (x) f (y), \forall x , y \in R and f (1) = 2. For a \DeltaABC, a = f (3), b = f (1) + f (3) and c = f (2) + f (3). If r, r1, r2, r3 radii of incircle and excircles according to their usual notations then Column - I Column - II (P) r (1) 5 \sqrt{7} (Q) r1 (2) 3 \sqrt{7} (R) r2 (3) \frac{15}{\sqrt{7}} (S) r3 (4) \sqrt{7}

Engineering

Mathematics

Circumradius Inradius and exradius

Question

Let f (x + y) = f (x) f (y), $\forall x, y \in R$ and f (1) = 2. For a $∆$ ABC, a = f (3), b = f (1) + f (3) and c = f (2) + f (3).

If r, r₁, r₂, r₃ radii of incircle and excircles according to their usual notations then

P → 1

Q → 2

R → 3

S → 4

P → 2

Q → 1

R → 4

S → 2

P → 4

Q → 3

R → 2

S → 1

P → 3

Q → 4

R → 1

S → 3

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Putting x = y = 1, f (2) = 2² = 4

x = 2, y = 1, f (3) = 2³ = 8

$∴$ a = 8, b = 10, c = 12

$& s = \frac{a + b + c}{2} = 15, s - a = 7,$

s – b = 5, s – c = 3

$∴ Δ = \sqrt{s (s - a) (s - b) (s - c)}$

$= 15 \sqrt{7}$

$∴ r = \frac{Δ}{s} = \sqrt{7}$

$r_{1} = \frac{Δ}{s - a} = \frac{15}{\sqrt{7}}$

$r_{2} = \frac{Δ}{s - b} = 3 \sqrt{7}$

$& r_{3} = \frac{Δ}{s - b} = 5 \sqrt{7} .$