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General Equation of Hyperbola

If the length...

Question

If the length of tangents from vertices to incircle are in $H . P .$ then $r_{1}, r_{2}, r_{3}$ are in:

A . P .

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G . P .

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H . P .

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A . G . P .

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Solution

The correct option is A $A . P .$

Three length of tangents are given by $s - a, s - b, s - c$ respectively.
Given, length of the tangents are in $H . P .$
$\Rightarrow s - a, s - b, s - c$ are in $H . P .$
From relations, we know $Δ = r_{1} (s - a), Δ = r_{2} (s - b), Δ = r_{3} (s - c)$
$\Rightarrow \frac{Δ}{r_{1}}, \frac{Δ}{r_{2}}, \frac{Δ}{r_{3}}$ are in $H . P .$
Hence, its reciprocals will be in $A . P .$
$\Rightarrow \frac{r_{1}}{Δ}, \frac{r_{2}}{Δ}, \frac{r_{3}}{Δ}$ are in $A . P .$
$∴ r_{1}, r_{2}, r_{3}$ are in $A . P .$

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