First slide

Formation of differential equations

Question

A curve y=f(x) has the property that if the tangent drawn at any point p(x,y) on the curve meets the co-ordinate axes at A and B, then P is the midpoint of AB. If y(1)=1 then y(2)

Moderate

A

1
B

2
C

$\frac{1}{2}$
D

$- 2$

Solution

Equation tangent at P is y-y=m(x-x)----(1)
$\begin{array}{l} It meets x -axes at A (x - \frac{y}{y_{1}}, 0) B (0, y - x y_{1}) \\ P -is the midpoint of \bar{AB} \Rightarrow x - \frac{y}{y_{1}} = 2 x \\ \Rightarrow ydx + xdy = 0 \Rightarrow xy = c^{2} \\ If x = y = 1 \Rightarrow x y = 1 \\ If x = 2 \to y = 1 / 2 \end{array}$

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