Question-214683

Question Number 214683 by MATHEMATICSAM last updated on 16/Dec/24

Commented by MATHEMATICSAM last updated on 16/Dec/24

Circles C1 and C2 have equal radii and are tangent to the same line XY. Circle C3 is tangent to C1 and C2. Find distance h, from the centre of C3 to line XY in terms of x and radii of the circles. x is the distance between the centres of C1 and C2.

$$\mathrm{Circles}\:\mathrm{C1}\:\mathrm{and}\:\mathrm{C2}\:\mathrm{have}\:\mathrm{equal}\:\mathrm{radii}\:\mathrm{and} \\ $$$$\mathrm{are}\:\mathrm{tangent}\:\mathrm{to}\:\mathrm{the}\:\mathrm{same}\:\mathrm{line}\:\mathrm{XY}.\:\mathrm{Circle} \\ $$$$\mathrm{C3}\:\mathrm{is}\:\mathrm{tangent}\:\mathrm{to}\:\mathrm{C1}\:\mathrm{and}\:\mathrm{C2}.\:\mathrm{Find} \\ $$$$\mathrm{distance}\:{h},\:\mathrm{from}\:\mathrm{the}\:\mathrm{centre}\:\mathrm{of}\:\mathrm{C3}\:\mathrm{to}\:\mathrm{line} \\ $$$$\mathrm{XY}\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:{x}\:\mathrm{and}\:\mathrm{radii}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circles}. \\ $$$${x}\:\mathrm{is}\:\mathrm{the}\:\mathrm{distance}\:\mathrm{between}\:\mathrm{the}\:\mathrm{centres}\:\mathrm{of} \\ $$$$\mathrm{C1}\:\mathrm{and}\:\mathrm{C2}. \\ $$

Answered by mr W last updated on 16/Dec/24

a=radius of C1 and C2 b=radius of C3 (h−a)^2 +((x/2))^2 =(a+b)^2 ⇒h=a+(√((a+b)^2 −(x^2 /4)))

$${a}={radius}\:{of}\:{C}\mathrm{1}\:{and}\:{C}\mathrm{2} \\ $$$${b}={radius}\:{of}\:{C}\mathrm{3} \\ $$$$\left({h}−{a}\right)^{\mathrm{2}} +\left(\frac{{x}}{\mathrm{2}}\right)^{\mathrm{2}} =\left({a}+{b}\right)^{\mathrm{2}} \\ $$$$\Rightarrow{h}={a}+\sqrt{\left({a}+{b}\right)^{\mathrm{2}} −\frac{{x}^{\mathrm{2}} }{\mathrm{4}}} \\ $$

Commented by MATHEMATICSAM last updated on 16/Dec/24

$$\left.\mathrm{Thanks}\:\mathrm{sir}\::\right) \\ $$

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Question-214683

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