If PSQ is a focal chord of the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1` agtb then harmonic mean of SP and SQ is

Options

A. `(b^2)/(a`
B. `(a^2)/(b)`
C. `(2b^2)/(a^2)`
D. `(2b^2)/(b)`

Answer

To find the harmonic mean of the lengths \( SP \) and \( SQ \) for the focal chord \( PSQ \) of the ellipse given by the equation:

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

where \( a \geq b \), we can follow these steps:

### Step 1: Understand the properties of the ellipse
The ellipse is centered at the origin (0, 0) with semi-major axis \( a \) and semi-minor axis \( b \). The foci of the ellipse are located at \( (c, 0) \) and \( (-c, 0) \), where \( c = \sqrt{a^2 - b^2} \).

### Step 2: Identify the focal chord
A focal chord is a line segment that passes through one of the foci of the ellipse. In this case, we can consider the focal chord passing through the focus at \( (c, 0) \).

### Step 3: Length of the lattice rectum
The length of the lattice rectum \( L \) of the ellipse is given by the formula:

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

### Step 4: Harmonic mean of \( SP \) and \( SQ \)
The harmonic mean \( HM \) of two numbers \( SP \) and \( SQ \) is given by the formula:

\[ HM = \frac{2 \cdot SP \cdot SQ}{SP + SQ} \]

However, for a focal chord, it is known that the harmonic mean of the lengths of the segments from the focus to the endpoints of the chord is equal to half the length of the lattice rectum.

### Step 5: Calculate the harmonic mean
Since we established that the harmonic mean of \( SP \) and \( SQ \) is half the length of the lattice rectum, we can write:

\[ HM(SP, SQ) = \frac{1}{2} \cdot L = \frac{1}{2} \cdot \frac{2b^2}{a} = \frac{b^2}{a} \]

### Conclusion
Thus, the harmonic mean of \( SP \) and \( SQ \) is:

\[ \boxed{\frac{b^2}{a}} \]