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

Answer (Detailed Solution Below)View answer ▾

Answer: Option B — parabolas

Detailed Solution

The rectangle at (x, y) has area $x y$ ; the part below the curve is $\int_{0}^{x} y d x$ .

For the ratio 1 : 2, $\int_{0}^{x} y d x = \frac{1}{3} x y$ . Differentiating: $y = \frac{1}{3} (y + x y')$ , so $x y' = 2 y$ .

$\frac{d y}{y} = \frac{2 d x}{x}$ gives $y = k x^{2}$ , a family of parabolas. (The case $\frac{2}{3} x y$ gives $y = 2 x y'$ , i.e. $y^{2} = k x$ , also parabolas.)

✓ Check

y = x^{2}

x = 3

: area below

= \int_{0}^{3} x^{2} d x = 9

, rectangle

= 3 \times 9 = 27

, so the two parts are 9 and 18.

⚠ Common trap

(D) a straight line through the origin always cuts the rectangle into two equal halves, ratio 1 : 1.

Quick tip

$\int_{0}^{x} y d x = \frac{1}{3} x y \Rightarrow x y^{'} = 2 y \Rightarrow y = k x^{2}$

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Differential Equations — Preparation

This question tests your understanding of Differential Equations, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

A differential equation is an equation involving a function and its derivatives. This topic covers the order and degree of a differential equation, forming a differential equation by eliminating arbitrary constants, and solving first-order equations. The main solution methods are variables separable, homogeneous equations (using the substitution y = vx), and linear equations of the form dy/dx + Py = Q, solved with the integrating factor e^(integral of P dx). Exam questions also include application problems on growth and decay, such as population growth, radioactive decay and Newton's law of cooling. Remember that the degree is defined only when the equation is a polynomial in its derivatives. Practise the differential equations questions below, each with a step-by-step solution.

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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

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