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
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2022 Q.5)**
**1. Governing Concept**:
- **Topic**: Geometry & Pythagorean Tangent-Radius Relations
- Let $R$ be the center of the circle of radius $r$.
- Line $PQ$ is tangent to the circle at point $Q$, making radius $RQ \perp PQ$.
- In right-angled triangle $\triangle RQP$, by the Pythagorean theorem:
$RP^2 = RQ^2 + PQ^2 \implies PQ^2 = RP^2 - RQ^2$
- Now, look at the squares constructed on the line segments:
- Square $PXWR$ has side length equal to $RP \implies \text{Area}(PXWR) = RP^2$.
- Square $RUVZ$ has side length equal to radius $r = RQ \implies \text{Area}(RUVZ) = RQ^2$.
- Square $SPQT$ has side length equal to tangent length $PQ \implies \text{Area}(SPQT) = PQ^2$.
- Substituting into Pythagoras' theorem:
$\text{Area}(SPQT) = \text{Area}(PXWR) - \text{Area}(RUVZ)$
**2. Option Analysis**:
- **Option (A) is INCORRECT**: Tangent length and hypotenuse cannot all be equal.
- **Option (B) [Area of SPQT = Area of PXWR - Area of RUVZ] is CORRECT**: Direct geometric application of $PQ^2 = RP^2 - RQ^2$.
- **Option (C) is INCORRECT**: Inverted Pythagorean terms.
- **Option (D) is INCORRECT**: Hypotenuse cannot be subtracted from radius.
**3. Final Answer**:
- **Correct Choice**: **B**
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