State whether the given statement is true or false.
In given figure, A be one point of intersection of two intersecting circles with center O and Q. The tangents at A to the circles meet the circles again in B and C respectively. AOPQ is a parallelogram, then AP = PC.

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True

False

answer is A.

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Detailed Solution

Given figure,
Let A be one point of intersection of two intersecting circle with centres O and Q . The tangent at A to the two circles meet the circles again at B and

Let A be one point of intersection of two intersecting circle with centres O and Q . The tangent at A to the two circles meet the circles again at B and

The angle made by the intersection of the radial line and tangent at the circle is

90 o

.
As the angle made by the intersection of the radial line and tangent at the circle is,

90 °, A Q ⊥ A B

.
Also, since opposite sides of the parallelogram are parallel, it can be seen that

A Q ‖ O P

Suppose QP intersects AC at M.
As

A Q ‖ O P

and

A Q ⊥ A B, Q M ⊥ A C

.
As the perpendicular from the center of a center to the chord bisects the chord, this gives AM = CM.
Therefore, QM is the perpendicular bisector of AC.
This implies PQ is the perpendicular bisector of AC.
In,

△ A B C

since PQ is the perpendicular bisector of ,AC, AM = CM.
Also, OP is the perpendicular bisector of AB, AP = PB.
Similarly, PA = PC.
Therefore, AP = AC.
It is verified that, AP = AC, and hence, the given statement is true.
Therefore, the correct option is 1.

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