The locus of the mid-points of chords of ellipse, the tangents at the extremities of which intersect at right angles.

Options

A. `((x^2)/(a^2)+(y^2)/(b^2))(a^2-b^2)=x^2+y^2`
B. `((x^2)/(a^2)+(y^2)/(b^2))(a^2+b^2)=x^2+y^2`
C. `((x^2)/(a^2)+(y^2)/(b^2))(a^2+b^2)=x^2-y^2`
D. None of these

Answer

To find the locus of the midpoints of chords of an ellipse, where the tangents at the extremities of these chords intersect at right angles, we can follow these steps:

### Step 1: Understand the ellipse and its properties
The standard form of the equation of an ellipse is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
where \(a\) and \(b\) are the semi-major and semi-minor axes, respectively.

### Step 2: Determine the director circle
The director circle of the ellipse is defined by the equation:
\[ x^2 + y^2 = a^2 + b^2 \]
This circle is important because it helps in understanding the conditions under which the tangents at the endpoints of the chords intersect at right angles.

### Step 3: Write the equation of the chord of contact
For a point \((x_1, y_1)\) on the ellipse, the chord of contact is given by:
\[ \frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1 \]
This equation represents the line that is tangent to the ellipse at the point \((x_1, y_1)\).

### Step 4: Midpoint of the chord
Let the midpoint of the chord be \((h, k)\). The chord of contact for the midpoint can be expressed as:
\[ \frac{hx}{a^2} + \frac{ky}{b^2} = 1 \]

### Step 5: Relate the conditions for perpendicular tangents
For the tangents at the endpoints of the chord to intersect at right angles, the point \((x_1, y_1)\) must lie on the director circle. Therefore, we can set up the equation:
\[ x_1^2 + y_1^2 = a^2 + b^2 \]

### Step 6: Substitute and derive the locus equation
From the equations of the chord of contact and the condition for the director circle, we can compare the equations. By substituting the expressions for \(x_1\) and \(y_1\) in terms of \(h\) and \(k\), we can derive:
\[ \frac{h^2}{a^2} + \frac{k^2}{b^2} = 1 \]
This represents the locus of the midpoints of the chords.

### Step 7: Final locus equation
The final equation representing the locus of the midpoints of the chords of the ellipse, where the tangents at the extremities intersect at right angles, is:
\[ \frac{x^2}{\frac{a^2}{2}} + \frac{y^2}{\frac{b^2}{2}} = 1 \]
This is an ellipse with semi-major axis \(\frac{a}{\sqrt{2}}\) and semi-minor axis \(\frac{b}{\sqrt{2}}\).