Question

Through any point (x, y) of a curve which passes through the origin, lines are drawn parallel to the coordinate axes. The curve, given that it divides the rectangle formed by the two lines and the axes into two areas, one of which is twice the other, represents a family of:

• A. Circles
• B. Parabolas
• C. Hyperbolas
• D. Straight lines

Answer 1

Answer: (B)

Parabolas

Answer 2

Area of OAPB= xy ; area OAPO =

̃ $\int _ { 0 } ^ { \mathrm { x } } \mathrm { f } ( \mathrm { x } ) \mathrm { dx }$ = $\frac { 2 } { 3 } x$ f(x)

Diff. w.r.t. 'x' to get f(x) =$\frac { 2 } { 3 }$f(x) + $\frac { 2 } { 3 }$x f ¢ (x) or f(x) = 2f ¢ (x)

x ̃ = 2 or ln |y|= 2ln|x| + c

̃ y = cx²