Sections Of A Cone

Step 1: Concept
Standard hyperbola has eccentricity .
Step 2: Analysis
Given , so and .
.
Step 3: Conclusion
.
Final Answer: (A)

Step 1: Concept
For , the focus is at .
Step 2: Analysis
.
Step 3: Conclusion
The focus is .
Final Answer: (A)

Step 1: Point Analysis
The given point is the focus of the parabola .
Step 2: Graphical Analysis
For any point on the axis of the parabola , if , only one normal can be drawn (the axis itself). Here , so . Since , only the axis is a normal.
Final Answer: (b)

Step 1: Intersection Points
Solve and . . .
Step 2: Midpoint x-coordinate
The x-coordinate of the midpoint is the average of the roots: .
Step 3: Midpoint y-coordinate
Substitute into the chord equation : . The midpoint is .
Final Answer: (b)

Step 1: Simplify the Conic Form
Combine the trig terms: where .
Step 2: Compare with Standard Form
. Standard polar form is .
Step 3: Identify Eccentricity
Comparing, we get .
Final Answer: (c)

Step 1: General form of a hyperbola.
The standard form of a hyperbola with a horizontal transverse axis is: The distance between the foci is , and the distance between the vertices is . We are told that the distance between the foci is double the distance between the vertices, so:

Step 2: Relationship between , , and .
For hyperbolas, we know that: Substitute into this equation: Simplifying:

Step 3: Use the given conjugate axis length.
We are told that the length of the conjugate axis is 6, so: Thus, .

Step 4: Substitute into the equation of the hyperbola.
Substitute and into the standard form of the hyperbola: From , we find that , so the equation becomes: Multiplying both sides by 9:

Step 5: Conclusion.
Thus, the equation of the hyperbola is , corresponding to option (C).

Step 1: General form of the parabola.
The general equation of a parabola with its vertex at and focus at is: For a vertical parabola, the equation will be of the form: where is the distance from the vertex to the focus.

Step 2: Use the given coordinates.
We are given the vertex and focus , so the distance between the vertex and focus is . Thus, the equation of the parabola is:

Step 3: Simplify the equation.
Rearranging this equation:

Step 4: Conclusion.
Thus, the equation of the parabola is , which corresponds to option (C).

Step 1: Rewrite the equation of the hyperbola.
We are given the equation of the hyperbola: Divide both sides by 144 to simplify the equation: This is the standard form of the equation of a hyperbola.

Step 2: Determine the conjugate axis.
The conjugate axis of a hyperbola is perpendicular to the transverse axis. Since the transverse axis is along the x-axis, the conjugate axis is along the y-axis.

Step 3: Use the given conjugate line.
We are told that the conjugate line is . For a hyperbola, the conjugate line is related to the equation of the hyperbola through the coordinates of the points on the conjugate axis.

Step 4: Find the relationship between and .
The equation implies: Substitute this into the equation of the hyperbola: Simplifying:

Step 5: Solve for the value of .
Multiply the entire equation by the least common denominator, 576: Simplifying: Thus, the conjugate axis is related by the equation .

Step 1: Understanding the parabola and its focal radii.
In a parabola, the focus is a fixed point from which distances to any point on the parabola are equal to the perpendicular distance from the directrix. The focal radius is the distance from the focus to a point on the parabola.

Step 2: Consider the geometry of the circle.
When a circle is constructed on the focal radii of the parabola as its diameter, we are essentially looking at a circle whose diameter is the distance from the focus to the point on the parabola along the tangent.

Step 3: Circle touches the tangent at the vertex.
The circle that uses the focal radii of the parabola as its diameter will touch the tangent at the vertex of the parabola. This is because the tangent at the vertex is perpendicular to the axis of symmetry of the parabola, and the circle touches it at exactly the point where the radius meets the vertex.

Step 4: Conclusion.
Thus, the correct answer is that the circle touches the tangent at the vertex, corresponding to option (B).

Step 1: Formula / Definition}

Step 2: Calculation / Simplification}


Let :



Check : , RHS = (undefined)
only.
Step 3: Final Answer

Step 1: Formula / Definition}

Step 2: Calculation / Simplification}
For equilateral triangle of side :





Step 3: Final Answer

Concept: Tangents at latus rectum endpoints form a symmetric quadrilateral.

Step 1: Identify ellipse parameters.


Step 2: Length of latus rectum.


Step 3: Use symmetry property.
Area formed by tangents at endpoints gives fixed result.

Step 4: Final result.
Conclusion:
Area = 9

Concept: Ellipse standard form gives and foci at .

Step 1: Convert to standard form.


Step 2: Find focal distance.
Foci:

Step 3: Use distance formula.
Conclusion:
Radius =

Concept: Product of distances using roots

Step 1: Substitute line in parabola:

Step 2:

Step 3: Roots = .

Step 4: Use distance relation with point .

Step 5: Simplify product: Conclusion:
Answer =

Concept: Parametric form of parabola

Step 1: Vertex = (0,0), point .

Step 2: Slope of :

Step 3: Perpendicular slope = .

Step 4: Equation of through and meet x-axis.

Step 5: Projection on axis simplifies to: Conclusion:
Answer =

Concept: Tangent in slope form for parabola

Step 1: Tangent to is

Step 2: Tangent to is

Step 3: Common tangent equate both:

Step 4:

Step 5: Value of is not simple integer, so no option matches.
Conclusion:
Answer = None of these

Concept:

Step 1: Rewrite.


Step 2: Compute.

Concept: Tangent to parabola : Here

Step 1: Point lies on tangent:

Step 2: Rearrange: Slopes satisfy:

Step 3: Given

Step 4: Using sum/product relations: