Question:
Find the equation(s) of the tangents to which pass through
a) the point (2,2)
b) the point (4,6).
Attempted Solution:
a) For :
Let be parametrized as .
What do I do from here? The points given are not points of tangency; they are points that the tangents pass through. So, do I have to set something equal and solve for ?
We need to take care as the point is not on the ellipse.
If you plot the question both answers can be read off the graph. The tangent lines must be horizontal and vertical. Note that
Is the equation in standard form.
To solve it algebraically we need to find a point on the ellipse e.g it must satisfy and
Multiplying out the bottom equation gives
but since it must be on the ellipse this gives
Now putting this back into the ellipse again gives
as you can check this gives and correctly gives the point as when one of the tangent lines occur.