Let a tangent be drawn to the ellipse x²/27 + y² = 1 at (3√3 cos θ, sin θ) where θ ∈ (0, π/2). Then

Q: Wait, why do we use $\theta$ in the parametric form instead of just solving for $y$ in terms of $x$?

Great question! While you could solve for $y$ and use implicit differentiation, the parametric form $P(3\sqrt{3} \cos \theta, \sin \theta)$ is specifically designed to simplify the algebra of the ellipse. It turns a messy radical expression into clean trigonometric functions, making the differentiation much more manageable.

Q: How do I know that setting the derivative to zero gives a minimum and not a maximum?

Excellent intuition. In this specific geometric context, as $\theta$ approaches $0$ or $\pi/2$, the tangent line becomes nearly vertical or horizontal, causing the intercepts to approach infinity. Since the function $S(\theta)$ is continuous and positive on the interval $(0, \pi/2)$ and goes to infinity at the boundaries, the stationary point we find must be the global minimum.

Q: Is there a faster way to solve this without calculus?

Yes! You can use the AM-GM inequality (Arithmetic Mean-Geometric Mean). Since $S(\theta) = 3\sqrt{3} \sec \theta + \csc \theta$, you can manipulate the terms to apply AM-GM, but calculus is the standard, robust method for JEE Advanced problems as it works for a wider variety of functions.

Question

Let a tangent be drawn to the ellipse $\frac{x^2}{27} + y^2 = 1$ at $(3\sqrt{3} \cos 	heta, \sin 	heta)$ where $	heta \in (0, \pi/2)$. Then the value of $	heta$ such that the sum of intercepts on axes made by this tangent is minimum is equal to:

Options:

(A) $\pi/8$

(B) $\pi/4$

(C) $\pi/6$

(D) $\pi/3$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing on the curve of an ellipse, specifically the one defined by:

$$\frac{x^2}{27} + y^2 = 1$$

You are at a point $P$ defined by the parameter $	heta$. As you slide along this curve, you draw a tangent line at every position, which carves out intercepts on the $x$ and $y$ axes. Our mission is to find the exact angle $	heta$ where the sum of these intercepts is minimized.

### The Tangent Equation

The standard equation for a tangent to an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ at a point $(a \cos 	heta, b \sin 	heta)$ is:

$$\frac{x \cos 	heta}{a} + \frac{y \sin 	heta}{b} = 1$$

For our ellipse, $a^2 = 27$ and $b^2 = 1$, which implies $a = 3\sqrt{3}$ and $b = 1$. Substituting our point $P(3\sqrt{3} \cos 	heta, \sin 	heta)$, the equation becomes:

$$\frac{x \cos 	heta}{3\sqrt{3}} + y \sin 	heta = 1$$

### The Intercepts

To find the $x$-intercept, we set $y = 0$. The equation simplifies to $\frac{x \cos 	heta}{3\sqrt{3}} = 1$, which gives us $x = 3\sqrt{3} \sec 	heta$.

Similarly, for the $y$-intercept, we set $x = 0$, leading to $y \sin 	heta = 1$, or $y = \csc 	heta$. We now define our sum function $S(	heta)$ as the total length of the intercept sum:

$$S(	heta) = 3\sqrt{3} \sec 	heta + \csc 	heta$$

### The Calculus of Optimization

To find the minimum, we differentiate $S(	heta)$ with respect to $	heta$:

$$\frac{dS}{d	heta} = 3\sqrt{3} \sec 	heta 	an 	heta - \csc 	heta \cot 	heta$$

Setting this derivative to zero, we obtain:

$$3\sqrt{3} \sec 	heta 	an 	heta = \csc 	heta \cot 	heta$$

By converting these trigonometric functions into sine and cosine, we get:

$$3\sqrt{3} \frac{\sin 	heta}{\cos^2 	heta} = \frac{\cos 	heta}{\sin^2 	heta}$$

Rearranging this expression, we find:

$$\frac{\sin^3 	heta}{\cos^3 	heta} = \frac{1}{3\sqrt{3}}$$

This simplifies to $	an^3 	heta = (\frac{1}{\sqrt{3}})^3$. Taking the cube root, we arrive at:

$$	an 	heta = \frac{1}{\sqrt{3}}$$

Since $	heta$ is in the first quadrant, we conclude that the optimal angle is:

$$\mathbf{	heta = \frac{\pi}{6}}$$

Animated Solution for Mathematics - Conic Sections: Let a tangent be drawn to the ellipse $\frac{x ^{2}}{27} + y^{2} = 1$ at $(33 cos θ, sin θ)$ where $θ \in (0, π /2)$ . Then the value of $θ$ such that the sum of intercepts on axes made by this tangent is minimum is equal to:

Select Answer:

Visualized Solution

Problem Setup

Equation of Tangent

Substituting Point $P$

Simplified Tangent Equation

Finding the $x$ -intercept

Finding the $y$ -intercept

Sum of Intercepts Function

Differentiating $S (θ)$

Condition for Minimum

Converting to Sine and Cosine

Solving for $tan θ$

Final Value of $θ$

The Sigma Insight: Equation of Tangent and Normal

Analyzing the Setup

The Tangent Equation

The Intercepts

The Calculus of Optimization

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