You are assuming that the point (27, 3) is on the ellipse when you use your formula for y', but this is not a point on the ellipse. I would approach this problem by sketching a graph of the ellipse, and drawing a line from the given point to that it is tangent to the ellipse at some unknown point (x0, y0). The slope at the tangent to the ellipse at this point has to be equal to what your derivative says, and also has to be equal to the slope of the line between this point and the point (27, 3).General_Sax said:Homework Statement
Find the equations of both the tangent lines to the ellipse x2 + 9y2 = 81 that pass through the point (27, 3).
One is horizontal the other is not.
Homework Equations
The Attempt at a Solution
horizontal, easy: y = 3
x^2+9y^2=81
derivative:
2x + 18yy` = 0
y`= -x/9y
at the point (27,3) the slope will be -1.
y-3 = -(x-27)
y= -x + 30
this solution is wrong according to my online assignment program, but I can't for the life of me see why.
A tangent line to an ellipse is a line that touches the ellipse at exactly one point, called the point of tangency. This line is perpendicular to the ellipse's curve at that point.
To find the equation of a tangent line to an ellipse, you will need to know the coordinates of the point of tangency. Then, you can use the point-slope form or the slope-intercept form to write the equation of the tangent line.
The slope of a tangent line to an ellipse is equal to the slope of the ellipse at the point of tangency. This is because the tangent line is perpendicular to the curve of the ellipse at that point, which means the slopes will be negative reciprocals of each other.
No, a tangent line can only intersect an ellipse at exactly one point. This is because the definition of a tangent line is that it touches the curve at exactly one point and is perpendicular to the curve at that point.
Yes, a tangent line to an ellipse can be horizontal or vertical. This occurs when the ellipse is a circle, as these lines are perpendicular to the curve at every point and only intersect at one point.