# Equation of an ellipse and tangents

• Kartik.
In summary, the equation of an ellipse with focus (f,g), directrix Ax+By+C=0, and eccentricity e is (x-f)^2+(y-g)^2 = e^2(Ax+By+C)^2 / A^2+B^2. The variables x and y do not refer to the same thing in the locus of the directrix and the ellipse. To find the points of intersection of tangents from a point outside the curve on an elliptical curve, simply use the original point.
Kartik.
1) If we have the focus as (f,g) and the directrix as Ax+By+C =0 and the eccentricity as e we define the equation of the ellipse to be

(x-f)2+(y-g)2 = e2(Ax+By+C)2 / A2+B2

Does this imply that the variables x and y in the locus of the directrix and the ellipse refer to the same thing?(we do take them as similar or say like variables)

2) How can we find the points of intersection of the tangents(from a point outside the curve) on an elliptical curve?

Hi Kartik!
Kartik. said:
1) If we have the focus as (f,g) and the directrix as Ax+By+C =0 and the eccentricity as e we define the equation of the ellipse to be

(x-f)2+(y-g)2 = e2(Ax+By+C)2 / A2+B2

Does this imply that the variables x and y in the locus of the directrix and the ellipse refer to the same thing?(we do take them as similar or say like variables)

No!

Ax+By+C = constant is a series of parallel lines that fill out the plane.

If the constant is zero, the line is the directrix, and (x,y,z) lies on the directrix.

If the constant isn't zero, that tells you how far away from the directrix the line (and (x,y,z) itself) is
2) How can we find the points of intersection of the tangents(from a point outside the curve) on an elliptical curve?

uhh?

isn't that just the original point?

## What is the equation of an ellipse?

The equation of an ellipse is a mathematical representation of an oval-shaped curve. It is written in the form (x-h)^2/a^2 + (y-k)^2/b^2 = 1, where (h,k) is the center of the ellipse, a is the length of the major axis, and b is the length of the minor axis.

## How do you find the equation of an ellipse?

To find the equation of an ellipse, you need to know the coordinates of the center, the lengths of the major and minor axes, and the orientation of the ellipse. You can also find the equation by using the distance formula and the definition of an ellipse.

## What are tangents of an ellipse?

Tangents of an ellipse are lines that touch the ellipse at exactly one point. They are perpendicular to the radius at the point of tangency and have the same slope as the tangent line at that point. These lines are important in understanding the shape and properties of an ellipse.

## How do you find the tangents of an ellipse?

The tangents of an ellipse can be found by using the slope-intercept form of a line and the point-slope form of a line. You can also use the derivative of the equation of an ellipse to find the equation of the tangent line at a specific point.

## What is the significance of tangents in the study of ellipses?

Tangents are important in the study of ellipses because they help us understand the relationship between the curve and its tangent lines. They also help us determine the slope of the curve at a specific point and the direction in which the curve is moving. Tangents are also used in many real-world applications, such as in engineering and physics.

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