Through a point $A$ on a circle, a chord $A P$ is drawn and on the tangent at $A$ a point $T$ is taken such that $A T = A P$ . If $T P$ produced meet the diameter through $A$ at $Q$ . Prove that the limiting value of $A Q$ when $P$ moves upto $A$ is double the diameter of the circle.

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Solution

$∠ A P T = ∠ A T P = θ$

$\Rightarrow ∠ A P Q = π - θ$

$\Rightarrow A Q P = \frac{π}{2} - θ$

In $△ A P Q$

$\Rightarrow \frac{A Q}{sin (π - θ)} = \frac{A P}{sin (\frac{π}{2} - θ)}$

$\Rightarrow A Q = tan θ . A P$ ... $(1)$

In right angle $△ A P L$

$\Rightarrow (cos 2 θ - \frac{π}{2}) = \frac{A P}{A L} = \frac{A P}{2 r}$

$\Rightarrow A P = 2 r sin θ$ ... $(2)$

When $P \to A$

$\Rightarrow A P = A T$

$\Rightarrow T \to A \Rightarrow θ \to \frac{π}{2}$

$\Rightarrow A Q = {lim}_{θ \to \frac{π}{2}} \frac{sin θ}{cos θ} \times 2 r \times 2 sin θ cos θ$

$\Rightarrow {lim}_{θ \to \frac{π}{2}} 4 r {sin}^{2} θ$

$\Rightarrow 4 r = 2 (d i a m e t e r)$

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Through a point A on a circle, a chord AP is drawn and on the tangent at A a point T is taken such that AT=AP. If TP produced meet the diameter through A at Q. Prove that the limiting value of AQ when P moves upto A is double the diameter of the circle.

Through a point $A$ on a circle, a chord $A P$ is drawn and on the tangent at $A$ a point $T$ is taken such that $A T = A P$ . If $T P$ produced meet the diameter through $A$ at $Q$ . Prove that the limiting value of $A Q$ when $P$ moves upto $A$ is double the diameter of the circle.