The curve for which the slope of the tangent at any point is equal to the ratio of the abscissa to the ordinat

NCERT Exemplar · Q84

Q. The curve for which the slope of the tangent at any point is equal to the ratio of the abscissa to the ordinate of the point is: (A) an ellipse (B) parabola (C) circle (D) rectangular hyperbola

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Concept understanding — Differential Equation Modeling

Differential Equation Modeling: From Intuition to Precision

Imagine you're watching a cup of hot coffee cool down on your desk. It starts hot, then gradually loses heat to the room. But here's the key question: how fast does it cool at any given moment? The answer isn't a fixed number — it depends on how hot the coffee is right now. The hotter it is, the faster it cools. This is the core idea behind differential equation modeling: the rate of change of a quantity depends on the quantity itself.

The Intuition First

Suppose the room is at $20^\circ$ C and your coffee starts at $90^\circ$ C. Newton's Law of Cooling says the rate at which the coffee cools is proportional to the temperature difference between the coffee and the room. So when the coffee is $90^\circ$ C, the difference is $70^\circ$ C and it cools fast; when it's $40^\circ$ C, the difference is only $20^\circ$ C and it cools slowly. The rate changes as the temperature changes.

If we let $T(t)$ be the temperature at time $t$ , then "rate of change of temperature" is $\frac{dT}{dt}$ , and the statement above becomes:

\frac{dT}{dt} = -k(T - 20)

The minus sign is because the temperature is decreasing; the constant $k$ depends on the cup, the liquid, etc. This is a differential equation — an equation that involves a function and its derivative.

The Precise Statement

A differential equation is any equation that contains an unknown function and one or more of its derivatives. The goal is to find the function itself (here, $T(t)$ ) that satisfies the equation.

\frac{dy}{dx} = f(x, y)

Modeling means taking a real-world situation and translating it into a differential equation:

Identify the quantity you want to study (population, temperature, drug concentration, voltage).
State the rate law in words: "The population grows at a rate proportional to its size."
Translate into math: $\frac{dP}{dt} = kP$ .
Add initial conditions: $P(0) = P_0$ (the starting value).

Why This Matters

Without differential equations, you'd have to guess the whole future behavior of a system. With them, you get a precise rule that governs every instant of change. The solution to $\frac{dP}{dt} = kP$ is $P(t) = P_0 e^{kt}$ — exponential growth. The solution to the cooling equation is $T(t) = 20 + 70 e^{-kt}$ — exponential decay toward room temperature.

Note

Most real-world models are not this simple. But every model starts the same way: observe how something changes, express that change in terms of the thing itself, and write it as a differential equation.

A Simple Example to Try

A bacteria culture doubles every hour. If you start with 100 bacteria:

Step 1: Quantity = population $P(t)$ . …

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The condition $\frac{dy}{dx} = \frac{x}{y}$ leads to the differential equation $y\,dy = x\,dx$ , which integrates to $x^2 - y^2 = C$ , the equation of a rectangular hyperbola. The correct option is (D).

We are told: the slope of the tangent at any point $(x, y)$ on the curve equals the ratio of the abscissa ( $x$ ) to the ordinate ( $y$ ) at that point. Slope of tangent is $\frac{dy}{dx}$ . So the condition is:

\frac{dy}{dx} = \frac{x}{y}

This is a first-order differential equation. The key is to recognise that it is separable — we can bring all $y$ terms to one side and all $x$ terms to the other.

Separate the variables. Multiply both sides by $y$ and by $dx$ :

y \, dy = x \, dx

Integrate both sides. The integration is straightforward:

\int y \, dy = \int x \, dx

\frac{y^2}{2} = \frac{x^2}{2} + C_1

Multiply through by 2:

y^2 = x^2 + 2C_1

Let $2C_1 = C$ (an arbitrary constant). Then:

y^2 - x^2 = C

Or equivalently:

x^2 - y^2 = -C

Since $C$ is arbitrary, the sign doesn't matter. The standard form is:

x^2 - y^2 = k

where $k$ is any constant (positive, negative, or zero).

Identify the curve. The equation $x^2 - y^2 = k$ is the equation of a rectangular hyperbola (also called an equilateral hyperbola). Its asymptotes are perpendicular lines (the coordinate axes rotated by $45^\circ$ ). When $k = 0$ , it degenerates into the pair of lines $y = \pm x$ . …

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Q.The curve for which the slope of the tangent at any point is equal to the ratio of the abscissa to the ordinate of the point is:
(A) an ellipse
(B) parabola
(C) circle
(D) rectangular hyperbola

Concept understanding — Differential Equation Modeling

Differential Equation Modeling: From Intuition to Precision

The Intuition First

The Precise Statement

Why This Matters

A Simple Example to Try

Method: Turn a geometric slope condition into a differential equation, then name the curve

Steps

Common Mistakes

Q.The curve for which the slope of the tangent at any point is equal to the ratio of the abscissa to the ordinate of the point is: (A) an ellipse (B) parabola (C) circle (D) rectangular hyperbola

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Q.The curve for which the slope of the tangent at any point is equal to the ratio of the abscissa to the ordinate of the point is:
(A) an ellipse
(B) parabola
(C) circle
(D) rectangular hyperbola

Unlock everything free for 14 days