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Examoogle • User • info@examoogle.com • EE-2024-8821
Chapter 1 of 12 • Page 1 of 248🔒 Protected PDF • Watermarked
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PQ হল দুটি বৃত্তের একটি সাধারণ স্পর্শক যার কেন্দ্র যথাক্রমে O1 এবং O2, যা একটি বিন্দু A-তে বাহ্যিকভাবে একে অপরের সাথে স্পর্শ করে। বৃত্তগুলির ব্যাসার্ধ যথাক্রমে 11 সেমি এবং 4 সেমি। PQ এর দৈর্ঘ্য নির্ণয় করুন।
211 সেমি
311 সেমি
411 সেমি
511 সেমি
411 সেমি
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).
Two externally touching circles have radii r1 = 11 cm and r2 = 4 cm.
L=2r1×r2
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).
Confusing the formula for direct common tangent (2r1r2) with the formula for transverse common tangent (2d2−(r1+r2)2).
Identify given parameters
The radii of the two circles are r1=11 cm and r2=4 cm. Since they touch externally, the distance between centers d=r1+r2=15 cm.
r1=11,r2=4
Apply Direct Common Tangent Formula
The length L of a direct common tangent for two circles touching externally is given by L=2r1×r2.
L=2r1×r2
Calculate the Result
Substitute the radii values into the formula: L=211×4=244. Simplifying the square root, 44=4×11=211. Thus, L=2×211=411 cm.
L=411
C is correct because the direct common tangent length calculated using 2r1r2 is 411 cm.
This concept is frequently tested alongside the distance between centers of intersecting circles or internal common tangents in competitive geometry.