#### To determine

**To check:**

Effect on point R as P is taken closer and closer to the axis.

#### Answer

When P is taken closer to axis, the point R approaches the midpoint of the OA.

#### Explanation

**1) Concept:**

Here we use Laws of Cosines to simplify given examples.

**2) Formula:**

Laws of cosine: Let a, b, c be sides of triangle and A, B, C are corresponding opposite angles, then a2=b2+c2-2bc cosA.

**3) Calculations:**

From figure, ∠OQR=∠OQP

Therefore ∆QOR is isosceles.

Hence, QR=RO=x

By using Laws of cosines,

x2=x2+r2-2rx cosθ

r2=2rxcosθ

x=r2cosθ

As P goes closer to axis, that is y goes to 0, then θ goes to 0 as

sinθ=yr=0r=0

Hence x goes near to r2, since

x= r2cosθ and limx →0cos x=1

That is x goes near to midpoint of OA as OA=r.

**Conclusion:**

When P is taken closer to axis the point R approaches the midpoint of the OA.