Coordinate Geometry

Step 1: Concept
Find foci of the ellipse, then determine the radius of the circle.
Step 2: Analysis of Ellipse
, .
.
Foci .
Step 3: Circle Radius
Centre .
Radius .
Step 4: Calculation
Equation: .
.
Final Answer:(C)

The point P(-5, 0) lies in a Cartesian coordinate system. In this system, a point is defined by an ordered pair (x, y) where x is the horizontal component and y is the vertical component. The quadrants are defined as follows:

In addition to the quadrants, points can lie on the axes:

For the point P(-5, 0):

Since the y-coordinate is 0, the point P(-5, 0) lies on the x-axis.

To find the equation of the curve, we start with the given differential equation:

We separate the variables to integrate:

Integrating both sides:

This results in:

Solving for , we multiply through by 2:

Now, we use the point (2, 2) to find the constant :

This simplifies to:

Solve for :

Substitute back into the equation:

Thus, the equation of the curve is:

If and are the vertices of a parallelogram , then we need to find the value of .

In a parallelogram, the diagonals bisect each other. Therefore, the midpoint of diagonal will be equal to the midpoint of diagonal

The midpoint of is calculated as follows:

The midpoint of is calculated as follows:

Equating the two midpoints, we have:

Solving for , we multiply both sides by 2:

Subtracting 2 from both sides gives:

Therefore, the value of is 4.

To find the fourth vertex of a rhombus given three vertices, we use the property that the diagonals of a rhombus bisect each other at right angles and are equal in length from center to vertex pairs. The given points are A(2,-1), B(3,4), and C(-2,3).

Firstly, we calculate the midpoint of the diagonal that can be defined by segment AC:

Midpoint of AC = ((2 + (-2))/2, (-1 + 3)/2) = (0/2, 2/2) = (0,1).

This midpoint must also be the midpoint of the diagonal BD. Suppose D is the fourth vertex. Let D be (x,y). Then the midpoint of BD is:

Midpoint of BD = ((3 + x)/2, (4 + y)/2).

Since these midpoints are the same, we equate them:

(3 + x)/2 = 0 and (4 + y)/2 = 1.

From (3 + x)/2 = 0, we solve for x:

x = -3.

From (4 + y)/2 = 1, we solve for y:

4 + y = 2.

y = -2.

Thus, the fourth vertex D is (-3,-2), which is the correct option.