an ellipse whose major axis is parallel to x- axis is such that the segmengts of focal chord are 1 and 3 units

To understand the properties of the ellipse you're describing, let's break down the information step by step. An ellipse with its major axis parallel to the x-axis can be represented by the standard equation:

The Standard Form of an Ellipse

The equation for an ellipse centered at the origin with the major axis along the x-axis is:

(x²/a²) + (y²/b²) = 1

Here, 'a' represents the semi-major axis, and 'b' represents the semi-minor axis. The distance from the center to each focus is given by the formula:

c = √(a² - b²)

where 'c' is the distance from the center to each focus of the ellipse.

Understanding Focal Chords

A focal chord is a line segment that passes through one of the foci of the ellipse and has its endpoints on the ellipse itself. The lengths of the segments of a focal chord can provide valuable information about the ellipse's dimensions.

Given Information

You mentioned that the segments of the focal chord are 1 unit and 3 units. This means that if we denote the lengths of the segments from the focus to the points on the ellipse as 'p' and 'q', we have:

• p = 1 unit
• q = 3 units

The total length of the focal chord is then:

p + q = 1 + 3 = 4 units

Using the Properties of Ellipses

For any ellipse, the product of the lengths of the segments of a focal chord is equal to the square of the semi-minor axis:

p * q = b²

Substituting the values we have:

1 * 3 = b²

Thus, we find:

b² = 3

From this, we can determine that:

b = √3

Finding the Semi-Major Axis

Now, we need to find the semi-major axis 'a'. We know that the total length of the focal chord is also related to the semi-major axis. The relationship is given by:

p + q = 2a

Substituting the known values:

4 = 2a

From this, we can solve for 'a':

a = 2

Calculating the Distance to the Foci

Now that we have both 'a' and 'b', we can find 'c', the distance from the center to the foci:

c = √(a² - b²)

Substituting the values we found:

c = √(2² - (√3)²)

c = √(4 - 3) = √1 = 1

Final Equation of the Ellipse

Now that we have all the necessary values, we can write the equation of the ellipse:

(x²/2²) + (y²/(√3)²) = 1

This simplifies to:

(x²/4) + (y²/3) = 1

In summary, the ellipse you described has a semi-major axis of 2 units and a semi-minor axis of √3 units, with the equation:

(x²/4) + (y²/3) = 1