In the given figure $AB$ is tangent at $A$ to the circle with centre at $O$; point $D$ is interior to circle and $DB$ intersects the circle at $C$. If $BC = DC = 3$, $OD = 2$ and $AB = 6$, then find the value of $[r]$ (where $r$ is the radius of circle and $[.]$ represent G.I.F)
Step-by-Step Solution
Key Concept: Apply power of a point theorem: for point B outside the circle, BA² = BC · BD. Combined with the tangent-radius perpendicularity condition OA ⊥ AB and distance constraint OD = 2, this determines the radius through the relationship r² + 36 = BC · BD = 3 · 6 = 18, yielding r² = -18 (invalid), requiring careful application of the power of point for interior point D and re-examination of the geometric configuration.
Using the formula $r^2 = \frac{2(OD^2 + OB^2) - BD^2}{4}$ and the constraint $(OB)^2 = (OA)^2 + (AB)^2 = r^2 + 36$, we derive $4r^2 = 2(r^2 + 36) - 28$. Simplifying gives $r^2 = 22$, so $r = \sqrt{22}$. However, the boxed answer shows $|r| = 4$, suggesting additional constraints or a specific configuration.
<div class="key-concept"><strong>Key Concept:</strong> Apply power of a point theorem: for point B outside the circle, BA² = BC · BD. Combined with the tangent-radius perpendicularity condition OA ⊥ AB and distance constraint OD = 2, this determines the radius through the relationship r² + 36 = BC · BD = 3 · 6 = 18, yielding r² = -18 (invalid), requiring careful application of the power of point for interior point D and re-examination of the geometric configuration.</div>
<div class="trap-box"><strong>Trap:</strong> A common error is applying power of point formula incorrectly by treating B as external when D is interior to the circle. Students often compute BA² = BC · BD = 18 directly, leading to r² = 36 - 18 = 18 or r = √18 ≈ 4.24, and then incorrectly take [r] = 4 without verifying consistency with OD = 2 constraint. The actual derivation requires using the correct configuration where the chord CD passes through interior point D, making power of point applied at B yield r² = 22, giving r = √22 ≈ 4.69, so [r] = 4.</div>
Correct Answer: 4