12th Class
Mathematics
Definite Integrals
Question Bank
Critical Thinking
question_answer
For which of the following values of m, the area of the region bounded by the curve \[y=x-{{x}^{2}}\] and the line \[y=mx\] equals\[\frac{9}{2}\] [IIT 1999]
A)\[-4\]
B)\[-2\]
C)2
D)4
Correct Answer:
B
Solution :
The equation of curve is \[y=x-{{x}^{2}}\] \[\Rightarrow {{x}^{2}}-x=-y\] Þ \[{{\left( x-\frac{1}{2} \right)}^{2}}=-\left( y-\frac{1}{4} \right)\]
This is a parabola whose vertex is \[\left( \frac{1}{2},\frac{1}{4} \right)\]
Hence point of intersection of the curve and the line \[x-{{x}^{2}}=mx\Rightarrow x(1-x-m)=0\]i.e.,\[x=0\]or\[x=1-m\]