Conic Sections

Given sides of rectangle are and Centre of circle and radius of circle

Equation of circle

Step 1: Write the equation of the hyperbola.
The standard form of the hyperbola is given as: This represents the equation of a hyperbola with center at the origin, and the coefficients for the directrix can be found using this standard equation.
Step 2: Identify the coefficients.
Comparing the equation of the hyperbola with the general form, we can determine the values of , , and . After simplifying, we find that .
Step 3: Conclusion.
The correct answer is (C) 81.

Step 1: Find the equation of the latus rectum.
The equation of the latus rectum of the parabola is , and the length of the latus rectum is .
Step 2: Calculate the area.
The area bounded by the parabola and the latus rectum in the first quadrant is given by: Evaluating this integral gives: Step 3: Conclusion.
The correct answer is (B) .

Step 1: Understanding the problem.
We are asked to find the area of the region bounded by the curve and the lines , , and the X-axis. The area can be found using the definite integral of the function between the limits and .

Step 2: Setting up the integral.
The area is given by the integral: Solving this integral, we obtain the value .

Step 3: Conclusion.
Thus, the area of the region is 428 sq. units, which makes option (A) the correct answer.

Step 1: Find the distance between focus and directrix.
Distance of focus from the directrix is

Step 2: Determine the focal length .
For a parabola, the distance between focus and directrix equals .

Step 3: Use the formula for length of latus rectum.


Step 4: Conclusion.

Step 1: Analyze the foci of the ellipse and hyperbola.
For the ellipse, the foci are given by , and for the hyperbola, the foci are given by . Set the two expressions for equal to each other because the foci coincide.
Step 2: Solve for .
After solving, we find that .
Step 3: Conclusion.
The correct answer is (D) 7.

Step 1: Use the formula for .
The formula for is:
Step 2: Finding and .
From , we use the identity to find: Similarly, from , we find:
Step 3: Applying the formula for .
Substituting the values of and into the formula: Simplifying the expression, we get:
Step 4: Conclusion.
Thus, , which makes option (D) the correct answer.

Step 1: General property of hyperbola.
For any point on the hyperbola, the distance from the point to the foci satisfies the equation: Step 2: Apply the given conditions.
Given the equation of the hyperbola , and the foci at and , we know the relationship for the product of the distances from a point on the hyperbola to the foci is: Step 3: Conclusion.
Thus, the value of is .

Step 1: Identify the points of discontinuity.
We are given the function . The function is undefined where the denominator is zero, i.e., when . Solving this, we find as the only point of discontinuity.

Step 2: Conclusion.
Thus, the function is discontinuous at exactly one point, corresponding to option (D).

Step 1: Use the condition for tangency.
For the line to be tangent to the curve at , both the point must satisfy the curve's equation and the slope of the curve at that point must match the slope of the line.
Step 2: Solve for and .
Substituting into the equation of the curve, we solve for and , and find that and .
Step 3: Conclusion.
The correct answer is (D) 2, 0.

Step 1: Understanding the circuit.
The circuit described consists of switches and , and we need to find its symbolic representation.

Step 2: Analyzing the logic gates.
The circuit involves a logical OR gate and AND gate, and the correct symbolic expression needs to be derived based on how the switches are connected.

Step 3: Conclusion.
The correct symbolic form of the circuit is , which makes option (D) the correct answer.

Step 1: Understanding the formula for the angle between two lines.
The angle between two lines can be found using the formula: where and are the direction ratios of the two lines.

Step 2: Finding the direction ratios of the lines.
From the given equations of the lines, the direction ratios of and are:
Step 3: Calculating the angle.
Now, calculate the dot product and the magnitudes of the vectors: Thus: The angle between the lines is , which makes option (A) the correct answer.

Step 1: Equation of the parabola.
The equation of a parabola symmetric about the y-axis with vertex at is of the form . Given that the point lies on the parabola, we substitute into the equation:
Step 2: Focal distance.
The focal distance is given by , and since , the focal distance is 5.
Step 3: Conclusion.
Thus, the focal distance is 5, corresponding to option (B).

Step 1: Find the parametric equations for the line.
The parametric equations of the line joining two points and are: Substitute the given points into these equations.
Step 2: Solve for the parameter .
We know that the -coordinate of the point is 2, so solve for from the equation for : Solving for :
Step 3: Use the value of to find .
Substitute into the equation for :
Step 4: Conclusion.
Thus, the -coordinate of the point is .

Step 1: Simplify Function

.
Step 2: Differentiate
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Step 3: Condition for Increase
.
This happens when .
Final Answer: (B)

Step 1: Formula
.
Given .
Step 2: Vectors
, .
.
, .
Step 3: Calculation
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Square both sides: .
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Final Answer: (C)

Step 1: Find Foci of Ellipse
For ellipse , .
.
Foci .
Step 2: Hyperbola Parameters
Foci of hyperbola are .
Given , so .
Step 3: Find for Hyperbola
.
Equation: .
Final Answer: (B)

Concept:
Continuity at : Step 1: Left-hand limit. Using , , :
Step 2: Equate with .
Step 3: Right-hand limit.
Step 4: Equate with . But continuity requires same value as LHS: Conclusion: Option (D)

Concept: For the parabola the latus rectum is the line passing through the focus and perpendicular to the axis of the parabola. Important properties:

• Focus:
• Equation of latus rectum:
• End points of latus rectum: and

The required area is the region between the parabola and the line from to .
Step 1: {Write the area using integration.}
Step 2: {Use symmetry of the curve.} Since the region is symmetric about the -axis,
Step 3: {Evaluate the integral.} Substitute the limits: