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Byju's Answer

Standard XII

Mathematics

Condition for Two Lines to Be Parallel

For fixed pos...

Question

For fixed positions of $B (z_{2})$ and $C (z_{3})$ all the bisectors (internal) of $∠ A$ will pass through a fixed point which is

H.M. of

z_{2}

and

z_{3}

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A.M. of

z_{2}

and

z_{3}

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G.M. of

z_{2}

and

z_{3}

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none of these

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Solution

The correct option is D G.M. of $z_{2}$ and $z_{3}$
From the given figure
$∠ B O D = 2 ∠ B A D = ∠ A$
$∠ C O D = 2 ∠ C A D = ∠ A$
Hence
$\frac{z_{4}}{z_{2}} = e^{i A}$
$\frac{z_{3}}{z_{4}} = e^{i A}$
Hence
$z_{4}^{2} = z_{2} . z_{3}$ ...(i)
Now since all the internal bisectors of A will pass through D.
Hence
The fixed point is the G.M of $z_{2}$ and $z_{3}$ ... from(i)

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