Analytical Geometry

Step 1: Understanding the Concept:
Asymptotes are lines that a curve approaches as it heads towards infinity. We can find horizontal asymptotes by taking the limit of the function as , and vertical asymptotes by finding the values of x for which the function approaches .
Step 2: Key Formula or Approach:
We first rearrange the equation to express y in terms of x (or vice versa) and then analyze its behavior at infinity. Given equation:
Expand the equation:
Step 3: Detailed Explanation:
1. Finding Horizontal Asymptotes:
To find horizontal asymptotes, we analyze the behavior of y as . Rearrange the equation to solve for : Now, take the limit as : So, as , , which means . The horizontal asymptotes are and .
2. Finding Vertical Asymptotes:
To find vertical asymptotes, we analyze the behavior of x as . We can also find the values of x for which the denominator in the expression for becomes zero, causing y to go to infinity. The denominator is . Setting it to zero: The vertical asymptotes are and .
Step 4: Final Answer:
The asymptotes of the curve are the lines , , , and . This can be written as .

Step 1: Understanding the Concept:
The general second-degree equation represents a conic section. Under a specific condition on its coefficients, this conic section degenerates into a pair of straight lines. This condition can be expressed using the determinant of a related matrix.
Step 2: Key Formula or Approach:
The general second-degree equation can be represented in matrix form using homogeneous coordinates. The condition for the equation to represent a pair of straight lines is that the determinant of the associated 3x3 symmetric matrix is zero.
The matrix is: The condition is .
Step 3: Detailed Explanation:
We need to compute the determinant of the matrix . For the equation to represent a pair of straight lines, this determinant must be zero. Step 4: Final Answer:
The necessary and sufficient condition for the general equation of the second degree to represent a pair of straight lines is .

Step 1: Understanding the Concept:
The nature of the conic section represented by the general second-degree equation is determined by the discriminant and the sign of .
If , it's a pair of straight lines.
If :
: Ellipse
: Parabola
: Hyperbola

Step 2: Detailed Explanation:

(A) : . . This indicates a Parabola. (A) matches (IV).
(B) : . . This indicates a Hyperbola. (B) matches (I).
(C) : . First check : . . Since , it represents a pair of straight lines. (C) matches (II).
(D) : . . This indicates an Ellipse. (D) matches (III).
Step 3: Final Answer:
The correct pairings are (A)-(IV), (B)-(I), (C)-(II), (D)-(III). This corresponds to option (B).

Step 1: Understanding the Concept:
This question requires matching various conic sections (mostly hyperbolas) with their properties like eccentricity and asymptotes. For a hyperbola or , the eccentricity is , and asymptotes are or , respectively. The OCR seems to have made errors. We will correct the options as we solve. Option (A) must have to match (III). Let's assume there are typos and solve based on the given properties.
Step 2: Detailed Explanation:
Let's analyze each item in List-I.
(A) : This is a vertical hyperbola. Here . Eccentricity . So, (A) matches (III).
(C) : Divide by 224: . This is a horizontal hyperbola. Here . Eccentricity . So, (C) matches (I).
(D) : Multiply by 9: . This is a horizontal hyperbola. Here . Eccentricity . So, (D) matches (II).
(B) : This is a rotated hyperbola. The asymptotes of are given by the same equation. The combined equation of asymptotes for the general conic is where k makes the equation a pair of lines. A simpler approach for gives the slopes of asymptotes. For this rotated hyperbola, finding asymptotes is complex. However, given the matching, we deduce (B) must match (IV).
Step 3: Final Answer:
The correct pairings are (A)-(III), (B)-(IV), (C)-(I), (D)-(II). This does not match any of the given answer keys. Rechecking the provided keys, Option 3 is (A)-(I), (B)-(IV), (C)-(II), (D)-(III). This is also incorrect. There is likely a significant error in the question or options. Based on our correct derivations, the matching is (A-III), (C-I), (D-II). Let's assume there's a typo in (A) such that it matches (I), etc. This question is flawed.