The question is from the topic circles. In this problem details about two perpendicular chords are given and we have to find out the radius of the circle. CAT Geometry questions are heavily tested in CAT exam. Make sure you master Geometry problems. Knowing chord intersection theorem helps.

Question 19: Two mutually perpendicular chords AB and CD intersect at P. AP = 4, PB = 6, CP = 3. Find radius of the circle.

- 31.25
^{(1/2)} - 37.5
^{(1/2)} - 26
^{(1/2)} - 52
^{(1/2)}

31.25

Starts Sat, April 27th, 2019

When 2 chords AB and CD intersect at P then AP * PB = CP * PD

Hence 4 * 6 = 3 * PD

Thus, PD = 8

Now AB = AP + PB = 10

And CD = CP + PD

Thus, CD = 11

Consider the circle with center O.

Drop a perpendicular from O to chord AB and CD.

This will bisect the chords at X and Y i.e AX=XB and CY = YD.

Here AX = AP + PX i.e 5 = 4 + PX

PX = 1

Similarly, since CD = 11, PY+CP+YD = 11,

=> PY = 11-3-5.5 = 2.5.

PY = OX and PX = OY.

So, PXOY will from a rectangle as seen in the figure.

Now consider the triangle BOX, it is a right triangle where OB is the radius.

XB = 5, OX = 2.5

Then OB = ( OX^{2} + XB^{2} )^{1/2} OB = 31.25^{1/2} Thus radius = 31.25^{1/2} The radius can also be found out using the triangle YOD.

The question is **"Find radius of the circle."**

Choice A is the correct answer.

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