Ellipse

Step 1: Concept
Standardize the equation to . Eccentricity .
Step 2: Analysis
. Here (major axis) and (minor axis). .
Step 3: Conclusion
.
Final Answer: (B)

Step 1: Concept
The standard form has its center at .
Step 2: Analysis
Comparing the given equation: and .
Step 3: Conclusion
The center is .
Final Answer: (B)

Step 1: Given Values
Distance between foci . Length of minor axis .
Step 2: Use Ellipse Identity
. .
Step 3: Calculate Eccentricity
.
Final Answer: (c)

Step 1: Concept
The standard form has its center at .
Step 2: Analysis
Comparing the given equation: and .
Step 3: Conclusion
The center is .
Final Answer: (B)

Step 1: Concept
The number of real tangents depends on whether the point is outside, on, or inside the ellipse.
Step 2: Analysis
Substitute (3, 5) into .
Step 3: Evaluation
. Since , the point lies outside the ellipse.
Step 4: Conclusion
From any point outside an ellipse, exactly two real tangents can be drawn.
Final Answer: (c)

Step 1: Equation of the ellipse and circle.
The equation of the ellipse is: The equation of the circle is:

Step 2: Calculate the slopes at the points of intersection.
At the points of intersection, we find the derivatives of both equations with respect to and . For the ellipse: Solving for gives: For the circle: Solving for gives:

Step 3: Angle of intersection.
The angle of intersection between two curves is given by: where and are the slopes of the two curves at the point of intersection. Substitute the values of and from the previous step: Simplifying this expression gives:

Step 4: Conclusion.
Thus, the angle of intersection is: The correct answer is option (B).

Step 1: Understand the geometry of the ellipse.
The standard form of an ellipse with its center at and axes parallel to the coordinate axes is: where is the length of the semi-major axis and is the length of the semi-minor axis. The foci of the ellipse are located at if the major axis is horizontal or at if the major axis is vertical.

Step 2: Analyze the given points.
We are given:
- The center of the ellipse at ,
- One vertex at ,
- One focus at . The distance between the center and the vertex is the length of the semi-major axis , and the distance between the center and the focus is the focal distance .

Step 3: Calculate and .
The distance between the center and the vertex is 2 units, so: The distance between the center and the focus is 1 unit, so:

Step 4: Use the relationship .
For an ellipse, the relationship between , , and is given by: Substitute and into this equation:

Step 5: Write the equation of the ellipse.
Thus, the equation of the ellipse is: Simplifying:

Step 6: Conclusion.
Thus, the equation of the ellipse is , corresponding to option (A).

Step 1: Formula / Definition}

Step 2: Calculation / Simplification}
Take :


are in AP.
Step 3: Final Answer

Step 1: Formula / Definition}

Step 2: Calculation / Simplification}
Expand functions around :





Coefficient of = evaluated at linear term
= coefficient is
Step 3: Final Answer

Step 1: Understanding the Concept:
Equation of tangent to ellipse with slope is .

Step 2: Detailed Explanation:
Here , , .
.
Equation: .
Taking positive: .
Intercepts: -intercept: .
-intercept: .
Area = .

Step 3: Final Answer:
Option (C) 24 sq units.

Concept: Given ellipse: Major axis is fixed is constant.

Step 1: Coordinates of latus rectum endpoints.
For ellipse:

Step 2: Eliminate parameter .


Step 3: Substitute into ellipse.
Multiply by : Conclusion:

Concept: The equation of a normal to the ellipse in terms of slope is: Alternatively, the condition for to be a normal is:

Step 1: Identify and .

Step 2: Apply condition.

Step 3: Rearrange. Multiply both sides by : Wait — careful: Multiply by : That gives , which is not matching option (B). The correct standard condition is: