Find The Condition That The Curves; ax2 + by2 = 1 And a ‘x2 + B’ y2 = 1 May Cut Each Other Orthogonally (at Right Angles).

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Find the condition that the curves; ax2 + by2 = 1 and a ‘x2 + b’ y2 = 1 may cut each other orthogonally (at right angles).


Correct option is

Condition for orthogonality implies that the tangents to the curves at the point of intersection are perpendicular. If (x0y0) is the point of intersection, and m1m2 are slopes of the tangents to the two curves a this point, them  

             m1m2 = –1.

Let us find the point of intersection. Solving the equations simultaneously, 

            ax2 + by2 – 1 = 0    

            a'x2 + b'y2 – 1 = 0   


⇒   the point of intersection (x0y0) is given by    


The slope of tangent to the curve ax2 + by2 = 1 is   


and the slope of tangent to the curve  


For orthogonality,   


Using the values of x0 and y0,    




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