The locus of mid points of parts in between axes and tangents of ellipse `x^2/a^2 + y^2/b^2 =1`will be

Options

A. `(a^(2))/(x^(2)) + (b^(2))/(y^(2)) =4`
B. `(x^(2))/(a^(2)) + (y^(2))/(b^(2)) =4`
C. `(x^(2))/(a^(2)) + (y^(2))/(b^(2)) =16`
D. `(x^(2))/(a^(2)) + (y^(2))/(b^(2)) =25`

Answer

To find the locus of midpoints of the segments between the axes and the tangents of the ellipse given by the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), we can follow these steps:

### Step 1: Write the equation of the tangent to the ellipse
The general equation of the tangent to the ellipse at a point can be written as:
\[ y = mx + \sqrt{a^2 m^2 + b^2} \]
where \( m \) is the slope of the tangent.

### Step 2: Find the x-intercept of the tangent
To find the x-intercept, set \( y = 0 \):
\[ 0 = mx + \sqrt{a^2 m^2 + b^2} \]
Solving for \( x \):
\[ mx = -\sqrt{a^2 m^2 + b^2} \implies x_1 = -\frac{\sqrt{a^2 m^2 + b^2}}{m} \]

### Step 3: Find the y-intercept of the tangent
To find the y-intercept, set \( x = 0 \):
\[ y = \sqrt{a^2 m^2 + b^2} \]
Thus, the y-intercept is:
\[ y_1 = \sqrt{a^2 m^2 + b^2} \]

### Step 4: Determine the midpoints of the intercepts
The midpoint \( (h, k) \) of the segment between the x-intercept \( (x_1, 0) \) and the y-intercept \( (0, y_1) \) is given by:
\[ h = \frac{x_1 + 0}{2} = \frac{-\sqrt{a^2 m^2 + b^2}}{2m} \]
\[ k = \frac{0 + y_1}{2} = \frac{\sqrt{a^2 m^2 + b^2}}{2} \]

### Step 5: Express \( h \) and \( k \) in terms of \( m \)
From the expressions for \( h \) and \( k \):
\[ h = -\frac{\sqrt{a^2 m^2 + b^2}}{2m} \]
\[ k = \frac{\sqrt{a^2 m^2 + b^2}}{2} \]

### Step 6: Eliminate \( m \)
To eliminate \( m \), we can express \( m \) in terms of \( h \) and \( k \):
From \( k \):
\[ \sqrt{a^2 m^2 + b^2} = 2k \]
Substituting this into the expression for \( h \):
\[ h = -\frac{2k}{2m} \implies h = -\frac{k}{m} \implies m = -\frac{k}{h} \]

### Step 7: Substitute \( m \) back into the equation
Substituting \( m \) back into the equation for \( k \):
\[ k = \frac{\sqrt{a^2 \left(-\frac{k}{h}\right)^2 + b^2}}{2} \]
Squaring both sides:
\[ 4k^2 = a^2 \frac{k^2}{h^2} + b^2 \]
Rearranging gives:
\[ 4k^2 - \frac{a^2 k^2}{h^2} - b^2 = 0 \]

### Step 8: Rearranging to find the locus
Dividing through by \( k^2 \):
\[ 4 - \frac{a^2}{h^2} - \frac{b^2}{k^2} = 0 \]
Rearranging gives:
\[ \frac{a^2}{h^2} + \frac{b^2}{k^2} = 4 \]

### Step 9: Replace \( h \) and \( k \) with \( x \) and \( y \)
Thus, the locus of midpoints can be expressed as:
\[ \frac{a^2}{x^2} + \frac{b^2}{y^2} = 4 \]

### Final Result
The locus of midpoints of the segments between the axes and the tangents of the ellipse is given by:
\[ \frac{a^2}{x^2} + \frac{b^2}{y^2} = 4 \]