Geometry

Step 1: General equation of a circle
The standard equation of a circle is: Since the required circle passes through the origin, substituting : Thus, the equation simplifies to:

Step 2: Condition for orthogonality
Two circles cut each other orthogonally if the condition: is satisfied, where and are the centres of the two given circles. For the first circle: Comparing with : For the second circle: Comparing:

Step 3: Apply orthogonality condition

Step 4: Find the required equation
Substituting and into:

Step 5: Conclusion
Thus, the correct answer is:

Step 1: Identify the Required Conditions

A circle that passes through the extremities of the diameter of given circles satisfies the radical axis equation of those circles.

Step 2: Find the Radical Axis

The given circles are:

1.

2.

3.

Subtracting Equation 1 from Equation 2:

Dividing by 2:

This represents the radical axis.

Similarly, subtracting Equation 1 from Equation 3:

Dividing by 2:

Solving these two equations:

Multiplying the second equation by 3:

Subtracting:

Substituting in :

Thus, the centre of the required circle is .

Step 3: Find the Equation of the Required Circle

Since the required circle must be of the form:

with centre , we get:

From the given equations, the radius conditions lead to .

Step 4: Conclusion

Thus, the correct answer is:

Step 1: Find the coordinates of the extremities of the latus rectum
The latus rectum of an ellipse is given by the coordinates: where is the focal distance: Thus, the coordinates of the extremities of the latus rectum with positive ordinate are:

Step 2: Use the given parabola equation
The extremities satisfy the equation of the given parabola: Substituting and :

Step 3: Conclusion
Thus, the points satisfy: Thus, the correct answer is:

Step 1: Find the Direction Ratios of the Line

The direction ratios of the line of intersection of two planes are given by:

where and are the normal vectors of the given planes.

For the first plane:

For the second plane:

Computing the Cross Product:

Expanding along the first row:

Evaluating the determinants:

Thus, the direction ratios are:

Step 2: Compute

Step 3: Conclusion

Thus, the correct answer is:

Step 1: Finding the Equation of the Common Chord
The given circles are: Subtracting the second equation from the first:

Step 2: Finding the Required Circle
A circle passing through the intersection of these two given circles is of the form: Substituting and simplifying using the condition that the center lies on the common chord, we derive:

Step 3: Conclusion
Thus, the final answer is:

Step 1: Equation of the Foot of the Perpendicular
Given the point and the plane equation: The equation of the line perpendicular to the plane passing through is: Substituting these into the plane equation: Expanding:

Step 2: Finding the Foot of the Perpendicular
Substituting into parametric equations: Thus, the foot of the perpendicular is:

Step 3: Ratio Calculation
Using the given equation , Thus, the ratio is:

Step 1: Understanding the condition given
For the curve , the derivative at is: For the curve , the derivative at is: Using the chain rule: Substituting the given conditions:
Step 2: Check the product of the slopes
If two curves are orthogonal, then the product of their slopes should be: Substituting values:

Step 3: Interpretation
Since the product of slopes is , it means that the tangent at to one curve is perpendicular to the tangent at to the other curve. This implies that the tangent to one curve is normal to the other.

Step 4: Conclusion
Thus, the correct answer is:

Step 1: Find the Points of Intersection
The given equations are: Substituting into the parabola equation: Rearranging: Solving for using the quadratic formula: Corresponding values:

Step 2: Compute the Area Enclosed
The enclosed area is given by: Evaluating the first integral: Evaluating the second integral: Thus,

Step 3: Conclusion
Thus, the correct answer is:

Step 1: Harmonic Conjugate Formula
The harmonic conjugate of with respect to the line segment joining and satisfies the section formula in harmonic division: where divides internally in the ratio and divides externally in the same ratio.

Step 2: Find the internal ratio of division
Using the section formula in 3D, we set: Equating coordinates with : Solving for : From the first equation: Substituting into the second equation: So, .

Step 3: Find
Using the external section formula:

Step 4: Compute

Step 5: Conclusion
Thus, the correct answer is:

Step 1: Equation of the Tangent and Normal
For the hyperbola: the equation of the tangent at a point is: Setting to find the X-intercept ( ): The equation of the normal at is: Setting to find the Y-intercept ( ):

Step 2: Midpoint of
The midpoint of is:

Step 3: Finding the Locus
Let and . Using , we get:

Step 4: Conclusion
Thus, the final answer is:

Step 1: Standard form of the ellipse
The equation of an ellipse centered at the origin with its major axis along the x-axis is: The foci of the ellipse are located at: where .

Step 2: Use the given foci distance
The total distance between the foci is given as , so: Solving for :

Step 3: Use the latus rectum formula
The length of the latus rectum of an ellipse is given by: It is given that the latus rectum is 4, so: Solving for :

Step 4: Solve for
From the foci relation: Substituting : Using , substitute: Rearrange: Factoring: Since , we take . Substituting into : Thus:

Step 5: Conclusion
Thus, the correct answer is:

Step 1: Find the coordinates of
For the parabola , the parametric coordinates are:

Step 2: Find the feet of perpendiculars and
- is the foot of the perpendicular from to the -axis: - is the foot of the perpendicular from to the -axis:

Step 3: Find the equation of line
The slope of line is: Equation of :

Step 4: Condition for tangency to
The equation of the tangent to at is: Substituting and : Rewriting: Comparing slopes:

Step 5: Conclusion
Thus, the correct answer is:

Step 1: Finding the Equation of the Common Chord
The given circles are: Subtracting the second equation from the first:

Step 2: Finding the Required Circle
A circle passing through the intersection of these two given circles is of the form: Substituting and simplifying using the condition that the center lies on the common chord, we derive:

Step 3: Conclusion
Thus, the final answer is: