PQ એ અનુક્રમે $ O_1 $ અને $ O_2 $ કેન્દ્રોવાળા બે વર્તુળોની સામાન્ય સીધી સ્પર્શક છે, જે એકબીજાને બિંદુ A પ

Question

PQ એ અનુક્રમે $ O_1 $ અને $ O_2 $ કેન્દ્રોવાળા બે વર્તુળોની સામાન્ય સીધી સ્પર્શક છે, જે એકબીજાને બિંદુ A પર બાહ્ય રીતે સ્પર્શે છે. વર્તુળોની ત્રિજ્યા અનુક્રમે 11 સેમી અને 4 સેમી છે. PQ ની લંબાઈ શોધો.

Examoogle · Accepted Answer

The correct answer is C: $ 4\sqrt{11} $ સેમી. For circles touching externally, the direct common tangent length is twice the geometric mean of the radii, simply 2 times the square root of (11 multiplied by 4).

Step-by-Step Solution & Concepts:
Shortcut Trick: For circles touching externally, the direct common tangent length is twice the geometric mean of the radii, simply 2 times the square root of (11 multiplied by 4).

Formula Used: $$L = 2\sqrt{r_1 \times r_2}$$

Given: Two externally touching circles have radii r1 = 11 cm and r2 = 4 cm.

Steps: Identify given parameters: The radii of the two circles are $r_1 = 11$ cm and $r_2 = 4$ cm. Since they touch externally, the distance between centers $d = r_1 + r_2 = 15$ cm.. Apply Direct Common Tangent Formula: The length $L$ of a direct common tangent for two circles touching externally is given by $L = 2\sqrt{r_1 \times r_2}$.. Calculate the Result: Substitute the radii values into the formula: $L = 2\sqrt{11 \times 4} = 2\sqrt{44}$. Simplifying the square root, $\sqrt{44} = \sqrt{4 \times 11} = 2\sqrt{11}$. Thus, $L = 2 \times 2\sqrt{11} = 4\sqrt{11}$ cm.

Answer: C is correct because the direct common tangent length calculated using $2\sqrt{r_1r_2}$ is $4\sqrt{11}$ cm.

Exam Tip: This concept is frequently tested alongside the distance between centers of intersecting circles or internal common tangents in competitive geometry.

Examoogle · Answer

Option A: $ 2\sqrt{11} $ સેમી

Examoogle · Answer

Option B: $ 3\sqrt{11} $ સેમી

Examoogle · Answer

Option D: $ 5\sqrt{11} $ સેમી

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