PQ అనేది వరుసగా $ O_1 $ మరియు $ O_2 $ కేంద్రాలుగా ఉన్న రెం... | Mathematics MCQ (తెలుగు) | Examoogle

MathematicsGeometry

PQ అనేది వరుసగా $O_1$ మరియు $O_2$ కేంద్రాలుగా ఉన్న రెండు వృత్తాలకు సాధారణ స్పర్శరేఖ, ఇవి ఒకదానికొకటి A వద్ద బాహ్యంగా తాకుతాయి. వృత్తాల వ్యాసార్థాలు వరుసగా 11 సెం.మీ మరియు 4 సెం.మీ. PQ పొడవును కనుగొనండి?

$2\sqrt{11}$ cm

$3\sqrt{11}$ cm

$4\sqrt{11}$ cm

$5\sqrt{11}$ cm

Correct Answer

📐 MA • Math Direct FormulaMathematicsGeometry

$4\sqrt{11}$ cm

Quick Summary:

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).

📐MAMath SolutionDirect Formula

📋 Given

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

🔢 Formula Used

L=2r1×r2L = 2\sqrt{r_1 \times r_2}L=2r1​×r2​​

⚡ Exam Hall Shortcut / Speed 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).

⚠️ Common Student Trap / Pitfall

Confusing the formula for direct common tangent ( $2\sqrt{r_1r_2}$ ) with the formula for transverse common tangent ( $2\sqrt{d^2 - (r_1+r_2)^2}$ ).

📊 Diagram / Illustration

🔢 Step-by-Step Solution

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.

r1=11,r2=4r_1 = 11, r_2 = 4r1​=11,r2​=4

Apply Direct Common Tangent Formula

The length $L$ of a direct common tangent for two circles touching externally is given by $L = 2\sqrt{r_1 \times r_2}$ .

L=2r1×r2L = 2\sqrt{r_1 \times r_2}L=2r1​×r2​​

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.

L=411L = 4\sqrt{11}L=411​

✅

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

Core Concepts Used

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Geometry of Circles Direct Common Tangent Properties of Tangents

💡 EXAM TIP

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

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