If PSP' is a focal chord of the ellipse `(x^(2))/(7)+(y^(2))/(9)=1` then `(SP.SP')/(SP+SP')`=

Options

A. `7/3`
B. `7/6`
C. `(9)/sqrt(7)`
D. `(7)/sqrt(3)`

Answer

To solve the problem, we need to find the value of \((SP \cdot SP') / (SP + SP')\) for the focal chord \(PSP'\) of the ellipse given by the equation:

\[ \frac{x^2}{7} + \frac{y^2}{9} = 1 \]

### Step 1: Identify the parameters of the ellipse
The standard form of the ellipse is given as:

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

From the equation, we can identify:
- \(a^2 = 7\) (thus \(a = \sqrt{7}\))
- \(b^2 = 9\) (thus \(b = 3\))

### Step 2: Find the length of the lattice rectum
The length of the lattice rectum \(L_r\) of an ellipse is given by the formula:

\[ L_r = \frac{2a^2}{b} \]

Substituting the values of \(a^2\) and \(b\):

\[ L_r = \frac{2 \cdot 7}{3} = \frac{14}{3} \]

### Step 3: Calculate \((SP \cdot SP') / (SP + SP')\)
We know that for a focal chord \(PSP'\), the relationship between \(SP\) and \(SP'\) can be expressed in terms of the harmonic mean:

\[ \frac{SP \cdot SP'}{SP + SP'} = \frac{1}{2} \times \text{(Harmonic Mean of } SP \text{ and } SP') \]

### Step 4: Use the property of the harmonic mean
The harmonic mean of \(SP\) and \(SP'\) for a focal chord is given by:

\[ \text{Harmonic Mean} = \frac{2 \cdot SP \cdot SP'}{SP + SP'} = \frac{L_r}{2} \]

Thus:

\[ \frac{SP \cdot SP'}{SP + SP'} = \frac{1}{2} \cdot \frac{L_r}{2} = \frac{L_r}{4} \]

### Step 5: Substitute the value of \(L_r\)
Now substituting \(L_r = \frac{14}{3}\):

\[ \frac{SP \cdot SP'}{SP + SP'} = \frac{1}{4} \cdot \frac{14}{3} = \frac{14}{12} = \frac{7}{6} \]

### Final Answer
Thus, the value of \(\frac{SP \cdot SP'}{SP + SP'}\) is:

\[ \frac{7}{6} \]