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The angle of intersection of t...

# The angle of intersection of the curves (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 and x^(2)+y^(2)=ab, is

Updated On: 27-06-2022

Text Solution

"tan"^(-1)((a-b)/(sqrt(ab)))"tan"^(-1)((a+b)/(sqrt(ab)))"tan"^(-1)((a-b)/(2sqrt(ab)))none of these

Solution : We have, <br> C_(1):(x^(2))/(a^(2))+(y^(2))/(b^(2))=1 " "...(i) and C_(2):x^(2)+y^(2)=ab " "...(ii) <br> Let P (x_(1),y_(1)) be the point of intersection of these curves. Then, at point P , we have <br>  ((dy)/(dx))_(C_(1))=-(b^(2)x_(1))/(a^(2)y_(1)) and, ((dy)/(dx))_(C_(2))=-(x_(1))/(y_(1)) <br> Let  theta  be the angle of intersection of (i) and (ii) at point P. Then, <br> tan theta =|(((dy)/(dx))_(C_(1))-((dy)/(dx))_(C_(2)))/(1+((dy)/(dx))_(C_(1))((dy)/(dx))_(C_(2)))| <br>  rArr tan theta =|((-b^(2)x_(1))/(a^(2)y_(1))+(x_(1))/(y_(1)))/(1+(b^(2)x_(1)^(2))/(a^(2)y_(1)^(2)))| <br>  rArr tan theta =((a^(2)-b^(2))x_(1)y_(1))/(a^(2)y_(1)^(2)+b^(2)x_(1)^(2))  <br>  rArr tan theta =(a^(2)-b^(2))/(a^(2)b^(2))xx sqrt((a^(3)b^(3))/((a+b)^(2))) [(" Solving (i) and (ii), we have ") , (x_(1)^(2)=(a^(2)b)/(a+b)"," y_(1)^(2)=(ab^(2))/(a+b))] <br>  rArr tan theta =((a-b)/(sqrt(ab)))
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