A) \[2xy={{c}^{2}}\]
B) \[xy+{{c}^{2}}=0\]
C) \[4{{x}^{2}}{{y}^{2}}=c\]
D) None of these
Correct Answer: A
Solution :
[a] Let the given straight line be the axis of coordinates and let the equation of the variable line be \[\frac{x}{a}+\frac{y}{b}=1\] This line cuts the coordinate axes at A(a, 0) and B(0, b). therefore, Area of \[\Delta AOB=\frac{1}{2}ab={{c}^{2}}\] Or \[ab=2{{c}^{2}}\] (i) If (h, k) are the coordinates of the middle point of AB, then \[h=\frac{a}{2}\] and \[k=\frac{b}{2}\] On eliminating a and b form (i) and (ii), we get \[2hk={{c}^{2}}\] Hence, the locus of (h, k) is \[2xy={{c}^{2}}\].
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