Question
The value of m for which the area of the triangle included between the axes and any tangent to the curve xm y=bm is constant is
Answer: Option B
:
B
∴xmy=bm
Taking logarithm
mlogex+logey=mlogeb∴mx+1ydydx=0∴dydx=−myx
Equation of tangent at (x,y) is
Y−y=−myx(X−x)xY−xy=−myX+mxy⇒myX=xY=xy(1+m)⇒Xx(1+m)m+xx(1+m)=1
∴ Area of triangle OAB
=12.OA.OB
=12∣∣x(1+m)m∣∣|y(1+m)|=|xy|(1+m)22|m|Form=1,=|xy|(4)2=2|xy|(∵xy=b)=2|b|= constant
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:
B
∴xmy=bm
Taking logarithm
mlogex+logey=mlogeb∴mx+1ydydx=0∴dydx=−myx
Equation of tangent at (x,y) is
Y−y=−myx(X−x)xY−xy=−myX+mxy⇒myX=xY=xy(1+m)⇒Xx(1+m)m+xx(1+m)=1
∴ Area of triangle OAB
=12.OA.OB
=12∣∣x(1+m)m∣∣|y(1+m)|=|xy|(1+m)22|m|Form=1,=|xy|(4)2=2|xy|(∵xy=b)=2|b|= constant
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