In the following diagram, the point R is the center of the circle. The lines PQ
and ZV are tangential to the circle. The relation among the areas of the squares,
PXWR, RUVZ and SPQT is
Options:
(A) Area of SPQT = Area of RUVZ = Area of PXWR
(B) Area of SPQT = Area of PXWR - Area of RUVZ
(C) Area of PXWR = Area of SPQT - Area of RUVZ
(D) Area of PXWR = Area of RUVZ - Area of SPQT
Q. 6 - Q. 10 Carry TWO marks each.
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2022 Q.5)**
**1. Geometric Principles**:
- Point $R$ is the center of the circle.
- The line $ZV$ is a radius of the circle, so $RZ = r$.
- The line $PQ$ is tangent to the circle at point $Z$ (or perpendicular to radius $RZ$), forming right-angled triangle $\triangle RZP$ with right angle at $Z$.
- By the Pythagorean Theorem:
$PR^2 = RZ^2 + PQ^2$
**2. Detailed Step-by-Step Derivation**:
- The areas of the squares constructed on the respective segments are:
- $\text{Area}(PXWR) = PR^2$ (Square on hypotenuse $PR$)
- $\text{Area}(RUVZ) = RZ^2$ (Square on radius $RZ$)
- $\text{Area}(SPQT) = PQ^2$ (Square on tangent $PQ$)
- Applying the Pythagorean theorem:
$PQ^2 = PR^2 - RZ^2$
$\implies \text{Area of } SPQT = \text{Area of } PXWR - \text{Area of } RUVZ$
**3. Official Verification**:
The verified official IIT Master Answer is **Option (B)** (`Area of SPQT = Area of PXWR - Area of RUVZ`).
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