Doesn't "equally inclined" to the x and y axes mean that the slopes of the tangent lines have to be 1 or -1?sooyong94 said:Homework Statement
Find the equation of the tangents to the ellipse 4x^2+9y^2 = 36 which are equally inclined to the x and y-axis.
Homework Equations
Quadratic discriminant
The Attempt at a Solution
First I substituted y=mx+c into the ellipse, and determined its discriminant, and got c^2 = 9m^2 + 4
An equally inclined tangent of an ellipse is a line that intersects the ellipse at two points, making equal angles with the tangent line at those points. This line divides the ellipse into two symmetrical halves.
An ellipse has an infinite number of equally inclined tangents, as there are an infinite number of possible angles that can be formed with the tangent line at any given point on the ellipse.
Equally inclined tangents in an ellipse are important because they help to define the symmetry of the shape. They also play a role in determining the orientation and position of the ellipse in relation to other objects or shapes.
The equation for an ellipse can be used to calculate the coordinates of points on the curve. By finding the slope of the tangent line at two points on the ellipse and setting those slopes equal to each other, the coordinates of the equally inclined tangent line can be determined.
Equally inclined tangents are also associated with the concept of conjugate diameters in an ellipse, which are two diameters that intersect at the center of the ellipse and form equal angles with the tangent lines at their points of intersection.