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?

Question

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?

Aman Tomar · Answer

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 · Answer

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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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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