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Question Number 47034 by 23kpratik last updated on 04/Nov/18
find angel between spheres x^2 +y^2 +z^2 =29, x^2 +y^2 +z^2 +4x−6y−8z−47=0 (4,−3,2)
$${find}\:{angel}\:{between}\:{spheres}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} =\mathrm{29},\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} +\mathrm{4}{x}−\mathrm{6}{y}−\mathrm{8}{z}−\mathrm{47}=\mathrm{0}\:\:\left(\mathrm{4},−\mathrm{3},\mathrm{2}\right) \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 04/Nov/18
point (4,−3,2) satisfy both eqn of sphere now let p_1 be the tangent plane though(4,−3,2) with respect to first sphere and similarly p_2 be the tangent plane with respect to 2nd sphere determinanation of eqn plane p_1 and plane p_2 and angle between p_1 and p_2 is the required angle first sphere eqn... x^2 +y^2 +z^2 =29 centre c_1 (0,0,0) radius r_1 ((√(29)) ) direction ratio bdtween (0,0,0) and (4,−3,2) is formula {(x_2 −x_1 ),(y_2 −y_1 ),(z_2 −z_1 )}=4,−3,2 so eqn of plane p_(1 ) is formula A(x−x_1 )+B(y−y_1 )+C(z−z_1 )=0 A,B,C are direction ratio plane p_1 is 4(x−4)−3(y+3)+2(z−2)=0 4x−3y+2z−16−9−4=0 4x−3y+2z=29 ←plane p_1 the 2nd sphre x^2 +y^2 +z^2 +4x−6y−8z−47=0 centre (−2,3,4) direction ratio between (4,−3,2) and(−2,3,4) (6,−6,−2) eqn of second plane p_2 tangent to second sphere is A(x−x_1 )+B(y−y_1 )+C(z−z_1 )=0 6(x−4)−6(y+3)−2(z−2)=0 6x−6y−2z−24−18+4=0 6x−6y−2z=38←plane p_2 let required angle is θ formula cosθ=((a_1 a_2 +b_1 b_2 +c_1 c_2 )/( (√(a_1 ^2 +b_1 ^2 +c_1 ^2 )) ×(√(a_2 ^2 +b_2 ^2 +c_2 ^2 )))) cosθ=(((6×4)+(−6×−3)+(−2×2))/( (√((4)^2 +(−3)^2 +(2)^2 )) ×(√((6)^2 +(−6)^2 +(−2)^2 )) )) cosθ=((38)/( (√(29)) ×(√(76)))) θ=cos^(−1) (((38)/( (√(29)) ×(√(76)))))
$${point}\:\left(\mathrm{4},−\mathrm{3},\mathrm{2}\right)\:{satisfy}\:{both}\:{eqn}\:{of}\:{sphere} \\ $$$${now}\:{let}\:{p}_{\mathrm{1}} {be}\:{the}\:{tangent}\:{plane}\:{though}\left(\mathrm{4},−\mathrm{3},\mathrm{2}\right) \\ $$$${with}\:{respect}\:{to}\:{first}\:{sphere}\:{and}\:{similarly} \\ $$$${p}_{\mathrm{2}} \:{be}\:{the}\:{tangent}\:{plane}\:{with}\:{respect}\:{to}\:\mathrm{2}{nd}\:{sphere} \\ $$$${determinanation}\:{of}\:{eqn}\:{plane}\:{p}_{\mathrm{1}} {and}\:{plane}\:{p}_{\mathrm{2}} \\ $$$${and}\:{angle}\:{between}\:{p}_{\mathrm{1}} {and}\:{p}_{\mathrm{2}} \:{is}\:{the}\:{required}\:{angle} \\ $$$${first}\:{sphere}\:{eqn}… \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} =\mathrm{29}\:\:\:\:{centre}\:{c}_{\mathrm{1}} \left(\mathrm{0},\mathrm{0},\mathrm{0}\right)\:\:\:{radius}\:{r}_{\mathrm{1}} \left(\sqrt{\mathrm{29}}\:\right) \\ $$$${direction}\:{ratio}\:{bdtween}\:\left(\mathrm{0},\mathrm{0},\mathrm{0}\right)\:{and}\:\left(\mathrm{4},−\mathrm{3},\mathrm{2}\right)\:{is} \\ $$$$\:\:{formula}\:\left\{\left({x}_{\mathrm{2}} −{x}_{\mathrm{1}} \right),\left({y}_{\mathrm{2}} −{y}_{\mathrm{1}} \right),\left({z}_{\mathrm{2}} −{z}_{\mathrm{1}} \right)\right\}=\mathrm{4},−\mathrm{3},\mathrm{2} \\ $$$${so}\:{eqn}\:{of}\:{plane}\:{p}_{\mathrm{1}\:} {is} \\ $$$${formula}\:{A}\left({x}−{x}_{\mathrm{1}} \right)+{B}\left({y}−{y}_{\mathrm{1}} \right)+{C}\left({z}−{z}_{\mathrm{1}} \right)=\mathrm{0} \\ $$$${A},{B},{C}\:\:{are}\:{direction}\:{ratio} \\ $$$${plane}\:{p}_{\mathrm{1}} \:{is}\:\mathrm{4}\left({x}−\mathrm{4}\right)−\mathrm{3}\left({y}+\mathrm{3}\right)+\mathrm{2}\left({z}−\mathrm{2}\right)=\mathrm{0} \\ $$$$\mathrm{4}{x}−\mathrm{3}{y}+\mathrm{2}{z}−\mathrm{16}−\mathrm{9}−\mathrm{4}=\mathrm{0} \\ $$$$\mathrm{4}{x}−\mathrm{3}{y}+\mathrm{2}{z}=\mathrm{29}\:\:\:\leftarrow{plane}\:{p}_{\mathrm{1}} \\ $$$${the}\:\mathrm{2}{nd}\:{sphre}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} +\mathrm{4}{x}−\mathrm{6}{y}−\mathrm{8}{z}−\mathrm{47}=\mathrm{0} \\ $$$${centre}\:\left(−\mathrm{2},\mathrm{3},\mathrm{4}\right) \\ $$$${direction}\:{ratio}\:{between}\:\left(\mathrm{4},−\mathrm{3},\mathrm{2}\right)\:{and}\left(−\mathrm{2},\mathrm{3},\mathrm{4}\right) \\ $$$$\left(\mathrm{6},−\mathrm{6},−\mathrm{2}\right) \\ $$$${eqn}\:{of}\:{second}\:{plane}\:{p}_{\mathrm{2}} {tangent}\:{to}\:{second}\:{sphere} \\ $$$${is}\:{A}\left({x}−{x}_{\mathrm{1}} \right)+{B}\left({y}−{y}_{\mathrm{1}} \right)+{C}\left({z}−{z}_{\mathrm{1}} \right)=\mathrm{0} \\ $$$$\mathrm{6}\left({x}−\mathrm{4}\right)−\mathrm{6}\left({y}+\mathrm{3}\right)−\mathrm{2}\left({z}−\mathrm{2}\right)=\mathrm{0} \\ $$$$\mathrm{6}{x}−\mathrm{6}{y}−\mathrm{2}{z}−\mathrm{24}−\mathrm{18}+\mathrm{4}=\mathrm{0} \\ $$$$\mathrm{6}{x}−\mathrm{6}{y}−\mathrm{2}{z}=\mathrm{38}\leftarrow{plane}\:{p}_{\mathrm{2}} \\ $$$${let}\:{required}\:{angle}\:{is}\:\theta \\ $$$${formula} \\ $$$${cos}\theta=\frac{{a}_{\mathrm{1}} {a}_{\mathrm{2}} +{b}_{\mathrm{1}} {b}_{\mathrm{2}} +{c}_{\mathrm{1}} {c}_{\mathrm{2}} }{\:\sqrt{{a}_{\mathrm{1}} ^{\mathrm{2}} +{b}_{\mathrm{1}} ^{\mathrm{2}} +{c}_{\mathrm{1}} ^{\mathrm{2}} }\:\:×\sqrt{{a}_{\mathrm{2}} ^{\mathrm{2}} +{b}_{\mathrm{2}} ^{\mathrm{2}} +{c}_{\mathrm{2}} ^{\mathrm{2}} }}\: \\ $$$${cos}\theta=\frac{\left(\mathrm{6}×\mathrm{4}\right)+\left(−\mathrm{6}×−\mathrm{3}\right)+\left(−\mathrm{2}×\mathrm{2}\right)}{\:\sqrt{\left(\mathrm{4}\right)^{\mathrm{2}} +\left(−\mathrm{3}\right)^{\mathrm{2}} +\left(\mathrm{2}\right)^{\mathrm{2}} }\:×\sqrt{\left(\mathrm{6}\right)^{\mathrm{2}} +\left(−\mathrm{6}\right)^{\mathrm{2}} +\left(−\mathrm{2}\right)^{\mathrm{2}} }\:} \\ $$$${cos}\theta=\frac{\mathrm{38}}{\:\sqrt{\mathrm{29}}\:×\sqrt{\mathrm{76}}} \\ $$$$\theta={cos}^{−\mathrm{1}} \left(\frac{\mathrm{38}}{\:\sqrt{\mathrm{29}}\:×\sqrt{\mathrm{76}}}\right) \\ $$$$ \\ $$$$ \\ $$

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