Find the locus of the intersection of tangents of ellipse if the sum of the ordinates of the points of contact be equal to b.

Answer

To solve the problem of finding the locus of the intersection of tangents to an ellipse given that the sum of the ordinates of the points of contact is equal to \( b \), we can follow these steps:

### Step 1: Understand the Ellipse Equation
The standard equation of an ellipse centered at the origin 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 Points of Contact
Let the points of contact of the tangents to the ellipse be \( (x_1, y_1) \) and \( (x_2, y_2) \). According to the problem, we have:
\[ y_1 + y_2 = b \]

### Step 3: Write the Tangent Equation
The equation of the tangent to the ellipse at the point \( (x_1, y_1) \) is given by:
\[ \frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1 \]
Similarly, the tangent at the point \( (x_2, y_2) \) is:
\[ \frac{xx_2}{a^2} + \frac{yy_2}{b^2} = 1 \]

### Step 4: Find the Intersection of the Tangents
To find the intersection of the two tangents, we solve the equations simultaneously. We can express the two tangent equations as:
1. \( \frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1 \) (1)
2. \( \frac{xx_2}{a^2} + \frac{yy_2}{b^2} = 1 \) (2)

### Step 5: Solve the System of Equations
From equation (1), we can express \( y \) in terms of \( x \):
\[ y = \frac{b^2}{y_1} \left(1 - \frac{xx_1}{a^2}\right) \]
Substituting this expression for \( y \) into equation (2) will allow us to find the coordinates of the intersection point.

### Step 6: Use the Condition on Ordinates
Using the condition \( y_1 + y_2 = b \), we can substitute \( y_2 = b - y_1 \) into the equations and simplify to find the relationship between \( x \) and \( y \).

### Step 7: Derive the Locus Equation
After simplification, we can find the locus of the intersection point, which will typically yield a linear equation in \( x \) and \( y \).

### Final Result
The locus of the intersection of the tangents will be a straight line, and we can express it in the form:
\[ y = \frac{b}{2} \]
This indicates that the locus is a horizontal line where the sum of the ordinates of the points of contact equals \( b \).