Tangent is drawn to ellipse x²/27 + y² = 1 at (3√3 cos θ, sin θ) (where θ ∈ (0, π/2)). Then the valu

Q: Why can't we just use AM-GM inequality here? The sum of two positive terms is minimized when they are equal, right?

That is a classic trap! AM-GM is powerful, but it only applies when the terms are independent or when their product is constant. Here, the terms $\sec \theta$ and $\csc \theta$ are linked by the same variable $\theta$. They are not independent, so you cannot simply equate them. Calculus is the robust path here.

Q: How do I know if the value I found is a minimum and not a maximum?

Great question! In JEE problems of this type, the function $S(\theta)$ tends to infinity as $\theta$ approaches $0$ or $\pi/2$. Since the function is continuous and differentiable in the interval $(0, \pi/2)$, the only critical point we find must correspond to the global minimum.

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

Yes, there is! You can use the property that for a tangent to an ellipse, the product of the intercepts is related to the area, or use the parametric form of the tangent and optimize using trigonometric inequalities. However, the calculus method is the most reliable and systematic approach for the exam.

Question

Tangent is drawn to 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 sum of intercepts on axes made by this tangent is minimum, is

Options:

(A) $\pi/3$

(B) $\pi/6$

(C) $\pi/8$

(D) $\pi/4$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

The ellipse is defined by the equation:

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

We are given a point $P$ on this curve with parametric coordinates $(3\sqrt{3} \cos 	heta, \sin 	heta)$. As $	heta$ varies in the first quadrant, the tangent line at $P$ creates intercepts on the coordinate axes. Our goal is to minimize the sum of these intercepts.

### The Geometry of the Tangent

The standard equation of a tangent to an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ at a point $(x_1, y_1)$ is given by:

$$\frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1$$

Substituting $a^2 = 27$, $b^2 = 1$, and the point $P(3\sqrt{3} \cos 	heta, \sin 	heta)$, we obtain:

$$\frac{x(3\sqrt{3} \cos 	heta)}{27} + \frac{y \sin 	heta}{1} = 1$$

Simplifying this expression, we arrive at the equation of the tangent line:

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

### The Intercepts: Defining the Function

The $x$-intercept $a$ occurs when $y=0$. Setting $y=0$ in our tangent equation, we find:

$$a = \frac{3\sqrt{3}}{\cos 	heta} = 3\sqrt{3} \sec 	heta$$

Similarly, the $y$-intercept $b$ occurs when $x=0$:

$$b = \frac{1}{\sin 	heta} = \csc 	heta$$

We define the sum of the intercepts as the function $S(	heta)$:

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

### 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 the derivative to zero to find the critical point:

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

Converting to sines and cosines, we have:

$$3\sqrt{3} \left( \frac{\sin 	heta}{\cos^2 	heta} ight) = \frac{\cos 	heta}{\sin^2 	heta}$$

### The Final Revelation

Cross-multiplying the terms leads to:

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

Dividing by $\cos^3 	heta$ and $3\sqrt{3}$, we obtain:

$$	an^3 	heta = \frac{1}{3\sqrt{3}} = \left( \frac{1}{\sqrt{3}} ight)^3$$

Taking the cube root of both sides, we find:

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

In the first quadrant, this corresponds to the angle **$	heta = \frac{\pi}{6}$** (or $30^\circ$). Thus, the sum of the intercepts is minimized when **$	heta = \frac{\pi}{6}$**.

Animated Solution for Mathematics - Conic Sections: Tangent is drawn to 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 sum of intercepts on axes made by this tangent is minimum, is

Select Answer:

Visualized Solution

The Ellipse and Parametric Point

Standard Equation of Tangent

Substituting Parametric Coordinates

Simplified Tangent Equation

Finding the Intercepts

The Sum of Intercepts Function

Differentiating the Sum Function

Critical Points for Minima

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 Geometry of the Tangent

The Intercepts

The Calculus of Optimization

The Final Revelation

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