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If two circles are such that the centre of one lies on the circumference of the other, then the ratio of the common chord of two circles to the radius of any of the circles is :
- a)√3 : 2
- b)√3: 1
- c)√5 : 1
- d)none of these
Correct answer is option 'B'. Can you explain this answer?
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1:b) 2:c) $\sqrt{2}$:d) $\frac{1}{\sqrt{2}}$
Answer: $\boxed{\textbf{(c) }\sqrt{2}}$
Solution: Let $O_1$ and $O_2$ be the centers of circles with radii $r_1$ and $r_2,$ respectively, and let $AB$ be the common chord. Without loss of generality, assume that circle $O_1$ is inside circle $O_2.$
[asy] unitsize(0.8cm); pair O1 = (0,0), O2 = (2,0); real r1 = 1.5, r2 = 2.5; pair A = (-1.5*sqrt(2),-1.5), B = (1.5*sqrt(2),-1.5); draw(Circle(O1,r1)); draw(Circle(O2,r2)); draw(A--B); label("$O_1$",O1,NW); label("$O_2$",O2,NW); dot("$A$",A,SW); dot("$B$",B,SE); label("$r_1$",O1--O1+dir(-90)*r1,SW); label("$r_2$",O2--O2+dir(-90)*r2,SW); draw(O1--A--O2--cycle,dashed); [/asy]
Since $O_1A=r_1$ and $O_2B=r_2,$ we have \[AB = O_1A + O_2B = r_1+r_2.\] By the Pythagorean Theorem, we have \[O_1O_2 = \sqrt{(O_1A)^2 + (O_2B)^2} = \sqrt{r_1^2 + r_2^2}.\] Then by the Pythagorean Theorem on right triangle $O_1O_2B,$ \[O_1B = \sqrt{O_1O_2^2 - O_2B^2} = \sqrt{r_1^2 + r_2^2 - r_2^2} = \sqrt{r_1^2} = r_1.\] Thus, \[AB = r_1 + r_2 = \sqrt{2}\cdot r_1,\] and the desired ratio is $\boxed{\sqrt{2}}.$
Answer: $\boxed{\textbf{(c) }\sqrt{2}}$
Solution: Let $O_1$ and $O_2$ be the centers of circles with radii $r_1$ and $r_2,$ respectively, and let $AB$ be the common chord. Without loss of generality, assume that circle $O_1$ is inside circle $O_2.$
[asy] unitsize(0.8cm); pair O1 = (0,0), O2 = (2,0); real r1 = 1.5, r2 = 2.5; pair A = (-1.5*sqrt(2),-1.5), B = (1.5*sqrt(2),-1.5); draw(Circle(O1,r1)); draw(Circle(O2,r2)); draw(A--B); label("$O_1$",O1,NW); label("$O_2$",O2,NW); dot("$A$",A,SW); dot("$B$",B,SE); label("$r_1$",O1--O1+dir(-90)*r1,SW); label("$r_2$",O2--O2+dir(-90)*r2,SW); draw(O1--A--O2--cycle,dashed); [/asy]
Since $O_1A=r_1$ and $O_2B=r_2,$ we have \[AB = O_1A + O_2B = r_1+r_2.\] By the Pythagorean Theorem, we have \[O_1O_2 = \sqrt{(O_1A)^2 + (O_2B)^2} = \sqrt{r_1^2 + r_2^2}.\] Then by the Pythagorean Theorem on right triangle $O_1O_2B,$ \[O_1B = \sqrt{O_1O_2^2 - O_2B^2} = \sqrt{r_1^2 + r_2^2 - r_2^2} = \sqrt{r_1^2} = r_1.\] Thus, \[AB = r_1 + r_2 = \sqrt{2}\cdot r_1,\] and the desired ratio is $\boxed{\sqrt{2}}.$
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If two circles are such that the centre of one lies on the circumference of the other, then the ratio of the commonchord of two circles to the radius of any of the circles is :a)√3: 2b)√3: 1c)√5: 1d)none of theseCorrect answer is option 'B'. Can you explain this answer?
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If two circles are such that the centre of one lies on the circumference of the other, then the ratio of the commonchord of two circles to the radius of any of the circles is :a)√3: 2b)√3: 1c)√5: 1d)none of theseCorrect answer is option 'B'. Can you explain this answer? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If two circles are such that the centre of one lies on the circumference of the other, then the ratio of the commonchord of two circles to the radius of any of the circles is :a)√3: 2b)√3: 1c)√5: 1d)none of theseCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If two circles are such that the centre of one lies on the circumference of the other, then the ratio of the commonchord of two circles to the radius of any of the circles is :a)√3: 2b)√3: 1c)√5: 1d)none of theseCorrect answer is option 'B'. Can you explain this answer?.
If two circles are such that the centre of one lies on the circumference of the other, then the ratio of the commonchord of two circles to the radius of any of the circles is :a)√3: 2b)√3: 1c)√5: 1d)none of theseCorrect answer is option 'B'. Can you explain this answer? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If two circles are such that the centre of one lies on the circumference of the other, then the ratio of the commonchord of two circles to the radius of any of the circles is :a)√3: 2b)√3: 1c)√5: 1d)none of theseCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If two circles are such that the centre of one lies on the circumference of the other, then the ratio of the commonchord of two circles to the radius of any of the circles is :a)√3: 2b)√3: 1c)√5: 1d)none of theseCorrect answer is option 'B'. Can you explain this answer?.
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