Diagram above shows a circle of radius , with two non-parallel chords and . When extended, these chords intersect outside the circle at point .
Find the distance correct up to 2 decimal places.
Note : Figure not drawn up to scale.
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Let ∣ P A ∣ = a and ∣ P C ∣ = b . Then by the Intersecting Secant theorem we have that
∣ P A ∣ ∣ P B ∣ = ∣ P C ∣ ∣ P D ∣ ⟹ a ( a + 8 ) = b ( b + 1 2 ) , (i).
Next, since A B C D is a cyclic quadrilateral we know that ∠ A B D + ∠ A C D = 1 8 0 ∘ . But as ∠ A C D + ∠ A C P = 1 8 0 ∘ as well we have that ∠ A B D = ∠ A C P . Similarly ∠ C D B = ∠ C A P . Thus triangles Δ P C A and Δ P B D are similar, in which case
∣ A C ∣ ∣ P A ∣ = ∣ D B ∣ ∣ P D ∣ ⟹ 3 a = 5 b + 1 2 ⟹ b + 1 2 = 3 5 a .
Combining this with the equation (i) gives us that
a 2 + 8 a = ( 3 5 a − 1 2 ) ( 3 5 a ) ⟹ 9 a 2 + 7 2 a = ( 5 a − 3 6 ) ( 5 a )
⟹ 9 a + 7 2 = 5 ( 5 a − 3 6 ) ⟹ 9 a + 7 2 = 2 5 a − 1 8 0
⟹ 1 6 a = 2 5 2 ⟹ a = 1 6 2 5 2 = 4 6 3 = 1 5 . 7 5 .