Tangent is drawn to the ellipse (x^2)/(27)+y^2=1 at (3 √(3) cosθ, sinθ), where θ∈(0, θ/ 2). Then

Tangent is drawn to the ellipse (x^2)/(27)+y^2=1 at (3 √(3)...

Tangent is drawn to the ellipse $\frac{x^{2}}{27}+y^{2}=1$ at $(3 \sqrt{3} \cos \theta, \sin \theta)$, where $\theta \in(0, \theta / 2)$. Then, the value of $\theta$ such that sum of intercepts on axes made by this tangent is minimum, is (A) $\frac{\pi}{3}$ (B) $\frac{\pi}{6}$ (C) $\frac{\pi}{8}$ (D) $\frac{\pi}{4}$

Tangent is drawn to the ellipse $\frac{x^{2}}{27}+y^{2}=1$ at $(3 \sqrt{3} \cos \theta, \sin \theta)$, where $\theta \in(0, \theta / 2)$. Then, the value
of $\theta$ such that sum of intercepts on axes made by this tangent is minimum, is
(A) $\frac{\pi}{3}$
(B) $\frac{\pi}{6}$
(C) $\frac{\pi}{8}$
(D) $\frac{\pi}{4}$

Key Concepts

Ellipse

An ellipse is a type of conic section defined by an equation that represents a set of points whose distances to two fixed points (the foci) add up to a constant. Its standard form, which involves the squares of the variables divided by the squares of the semi-axes, provides a way to identify its shape and dimensions. Properties such as symmetry, the positions of its vertices, and the behavior of its tangents are essential for solving geometric problems involving ellipses.

Tangent Lines to Conic Sections

A tangent line to a conic section, such as an ellipse, is a line that touches the curve at exactly one point without crossing it. The tangent’s equation can be derived using methods like implicit differentiation or by applying the point-slope condition directly to the ellipse’s equation. Understanding tangent lines is critical when analyzing geometric features like intercepts, angles, and distances in problems involving curves.

Line Intercepts

Line intercepts refer to the points where a line crosses the coordinate axes, namely the x-intercept and the y-intercept. These values are often used to express the line in intercept form. In optimization problems, the sum or other combinations of these intercepts can form the objective function that needs to be minimized or maximized, making it a central concept in analytical geometry.

Optimization via Differentiation

Optimization via differentiation is a mathematical technique used to find the minimum or maximum values of a function by taking its derivative and setting it equal to zero to locate critical points. This method is widely employed in geometric problems where a condition needs to be optimized, such as minimizing the sum of certain distances or intercepts formed by tangents to curves.

Transcript

00:01 In discussion given tangent is drawn through the ellipse x square upon 27 plus y square equal to 1 at 3 times square root 3 times cos theta to sign theta where theta element of 0 02 theta upon 2.

00:34 So here then the value of theta such that some of intercept on axis made by this tangent is minimum.

00:43 So here the ellipse equation can be written as x square upon 27 plus y square upon 1 equal to 1.

00:58 So here we can get the equation of tangent.

00:59 So here we can get the equation of tangent.

01:01 So here equation of tangent is x times x sub 1 upon a square plus y y sub 1 upon b square equal to 1 so now we apply here you can see that x of 1 value is 3 times square 2 times x x 2 2 times cost theta and y sub 1 the value is sine theta so now we apply that formula here so we get x times cost theta upon three times square root 3 plus y times sine theta upon sine theta upon 1 equal to 1 though sum of intercept equal to here you can see that three times square root three times sec theta plus cosec theta equal to f of theta for the minimum we find the derivative of f of theta and that is the equal to zero so here for minimum derivative of theta equal to zero so now you can see that the derivative of f of theta equal three times square root three times sec x times 10x positive times negative cossack x times quad x so now we simplify this and here we get the derivative of f of theta equal to three times square root three times sine cube theta negative code cube theta upon sine square theta times course square theta so now here for minimum derivative of theta equal to zero so now we can see three times square root 3, sine cube theta negative cos cube theta equal to 0...

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