Each question contains Statement- 1 and Statement- 2 and has...

Each question contains Statement- 1 and Statement- 2 and has the following choices (a), (b), (c) and (d), out of which ONLY ONE is correct. (a) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement-1 (b) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement-1 (c) Statement- 1 is True, Statement- 2 is False (d) Statement- 1 is False, Statement- 2 is True Statement 1 The foci of the ellipse $\frac{(x-2)^{2}}{25}+\frac{(y-3)^{2}}{49}=1$ are at $(2,3+2 \sqrt{6})$ and $(2,3-2 \sqrt{6})$ and Statement 2 The foci of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ where $a>b$ are at $(a e, 0)$ and $(-a e, 0)$

Each question contains Statement- 1 and Statement- 2 and has the following choices (a), (b), (c) and (d), out of which ONLY ONE is correct.
(a) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement-1
(b) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement-1
(c) Statement- 1 is True, Statement- 2 is False
(d) Statement- 1 is False, Statement- 2 is True
Statement 1 The foci of the ellipse $\frac{(x-2)^{2}}{25}+\frac{(y-3)^{2}}{49}=1$ are at $(2,3+2 \sqrt{6})$ and $(2,3-2 \sqrt{6})$
and
Statement 2 The foci of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ where $a>b$ are at $(a e, 0)$ and $(-a e, 0)$

Key Concepts

Standard Form of an Ellipse

The standard form of an ellipse's equation typically appears as ((x-h)²/a²) + ((y-k)²/b²) = 1, where (h, k) is the center of the ellipse and a and b are the lengths of the semi-major and semi-minor axes, respectively. Understanding this form is crucial because it helps identify the orientation of the ellipse and its key features such as axes lengths and center location.

Determining the Foci of an Ellipse

The foci of an ellipse are special points located along its major axis, and their positions depend on the relationship between the lengths of the semi-axes. For an ellipse centered at (h, k), if a > b, the distance c from the center to each focus is found using the formula c² = a² - b². Recognizing and computing this value is essential in understanding the ellipse's geometric properties.

Translation of Conic Sections

Translation in the context of conic sections means shifting the center of the conic from the origin to another point (h, k). This shift affects the equation by replacing x and y with (x-h) and (y-k) respectively, while the fundamental properties, like the positions of the foci relative to the center, remain determined by the same formulas. This concept is important when analyzing ellipses that are not centered at the origin.

Interpretation of Mathematical Statements

Evaluating paired statements requires understanding not only the underlying mathematical concepts but also the logical connection between the content of the statements. This involves verifying the accuracy of mathematical results and determining whether a provided explanation correctly justifies a given statement, an essential skill in questions that compare and contrast multiple assertions.

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