Question

the given figure shows two parallel tangents xy and x'y' at the points a and b respectively to the circle with cetre o. another tangent cd is drawn paralel to ab at the point p. show that a circle with p as the centre and op as radius will pass through c and d .

Answer 1

Answer:

Step-by-step explanation: it is given that tangents XY and X’Y’ are parallel.

So, AC || BD

We know, rangers is perpendicular to the radius at the point of contact.

So, angle OAC = angle OBD = 90

in quadrilateral ACDB,

AB|| CD

AC || BD

Angle BAC= 90

Hence, it is clear that  Quadrilateral ACDB is a rectangle

So, angle C = angle D = 90

Now in quadrilateral OACP,

Angle A = 90

Angle C =90

Angle OPC = 90 and angle AOP = 90

This implies quadrilateral OACP is a rectangle.

So, OA = PC [ we know opposite sides are equal of rectangle]

similarly, quadrilateral OBDP is a rectangle

This implies OB = PD

Also OA = OB = OP [ theses are the radii of the same circles ]

So, PC = PD = OP

This implies , points O, C and D are equidistant from point ap

Hence, A circle with centre P and radius OP will pass through C and D

Answer 2

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