Ellipse Geometry

To find the length of the chord of the ellipse given by the equation , whose mid-point is , we need to use the chord of contact formula for ellipses.

1. Given the standard ellipse equation where and .
2. The equation of the chord with mid-point for this ellipse is Substituting and , we have: Simplifying, this becomes: Multiplying through by 4 gives:
3. The length of the chord of the ellipse can be calculated using the formula , where for the line equation in the form .
4. Here we have , i.e., , . Thus, , giving or due to the symmetry of trigonometric functions.
5. Now, calculate and : - If , then and . - If , then and .
6. Using either value, compute the length of the chord:
7. To find the actual length of the chord, consider the factor from it being scaled in the ellipse coefficients: Finally multiplying and simplifying gives .

Hence, the length of the chord is , so the correct answer is

.

Step 1: Understanding the Concept:
This problem involves logarithmic identities and base change properties. We must also consider the constraints for the existence of logarithms: the base must be positive and not equal to 1, and the argument must be positive.
Step 2: Key Formula or Approach:
1. Factorize the arguments: and .
2. Use and .
Step 3: Detailed Explanation:
The equation is: Using log properties: Since : Let : Multiplying by : Case 1: . Check Constraints: Base . Logarithm base must be . So, is rejected. Case 2: Check Constraints: For , base . Valid. For , base . Rejected. The only real solution is .
Step 4: Final Answer:
Sum of squares .

Given:
Point lies on ellipse and eccentricity

Step 1: Use eccentricity relation
For ellipse (major axis along -axis since ): Squaring:

Step 2: Use point (4,3)
Substitute : Then,

Step 3: Length of latus rectum
For ellipse: Rationalizing:

Final Answer:

Step 1: Understanding the Question:
We are given information about an ellipse (a focus and eccentricity) and a point that lies on it. We need to find the area of a triangle formed by this point, the given focus, and the origin.
Step 2: Key Formula or Approach:
1. The standard equation for an ellipse centered at the origin with foci on the x-axis is (with ).
2. The coordinates of the foci are .
3. The relationship between and is .
4. The area of a triangle with vertices and is given by . A simpler method is (1/2) * base * height if the base is along an axis.
Step 3: Detailed Explanation:
First, let's find the parameters and of the ellipse.
The focus is given as S(4, 0). For a standard ellipse, the foci are at .
So, .
The eccentricity is given as .
Substituting the value of :
.
Now, find using the relation .
.
So, the equation of the ellipse E is:
The point P(3, ) lies on the ellipse. So, its coordinates must satisfy the ellipse's equation.
Substitute and :
Now we need to find the area of the triangle POS. The vertices are:
P =
O =
S =
We can use the formula Area = .
Let's take the segment OS as the base of the triangle. This segment lies on the x-axis.
Base length = Distance between O(0,0) and S(4,0) = 4.
The height of the triangle corresponding to this base is the perpendicular distance from point P to the x-axis, which is simply the absolute value of the y-coordinate of P.
Height = .
Area of POS =
Area = .
Step 4: Final Answer:
The area of POS is .