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The circle ω touches the circle Ω internally at P . The centre O of Ω is outside ω . Let XY be a diameter of Ω which is also tangent to ω . Assume PY > PX . Let PY intersect ω at Z . If YZ = 2 PZ , what is the magnitude of angle PYX in degrees?

The circle ω touches the circle Ω internally at P. The centre O of Ω is outside ω. Let XY be a diameter of Ω which is also tangent to ω. Assume PY > PX. Let PY  intersect ω at Z. If YZ = 2PZ, what is the magnitude of angle PYX in degrees?

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Grade:9

2 Answers

Aman Tomar
17 Points
one year ago
 we have drawn the common tangent TPR ro the two given circles.
Let PZ=x so that YZ=2x.
XY is tangent to the smaller circle at Q.
But
Now, YQ² =YZ×YP=2x×3x=6x² or
YQ=√6× X.
Now, we apply sine rule in triangle YPQ whereby, sin(YQP)/sin(45°) =3X/(√6×X) =√3/2.
Clearly,
Aman Tomar
17 Points
one year ago
we have drawn the common tangent TPR ro the two given circles.
Let PZ=x so that YZ=2x.
XY is tangent to the smaller circle at Q.
once again by the aforesaid theorem. Moreover,
Now, YQ² =YZ×YP=2x×3x=6x² or
 YQ=√6x.
Now, we apply sine rule in triangle YPQ whereby, sin(YQP)/sin(45°) =3x/(√6x) =√3/2.
Clearly,
hence
Thus,

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