I would like to know if there's a way to calculate the a, b and c dashed lines in the image below. The angles should be 45 degrees for both.
Sorry if the drawing is a little uneven but the ellipse is suppose to be symmetrical along the x or y axis. I want to know if there's a way to calculate a, b and c (the dashed lines).
To the OP: I am providing my response as an answer only because my response is long-winded. Therefore, it will be easier for you to understand as an answer. This response is really a guide for you to follow to attack the problem + get help on mathSE.
Review https://en.wikipedia.org/wiki/Ellipse.
Assuming that the minor (vertical) axis has length 2 x 6.82 = 13.64
and the major (horizontal) axis is 2 x 9 = 18
set up the equation for the ellipse in terms of x and y.
Resolve the following ambiguities in the problem:
Near the bottom of the diagram, does the vertical axis intersect the top of the rectangle in the middle of the upper side of the rectangle? That is, does the vertical axis bisect the width of the upper side of the rectangle? If not, where does the vertical axis intersect the upper side of the rectangle.
You have a line segment labeled "a" that intersects the ellipse. Does this line segment bisect the upper right arc of the ellipse? If not, where do you intend that this line segment intersect the ellipse?
You have a line segment labeled "b" that intersects the long side of the rectangle. Where along this side do you intend that the line segment intersect the rectangle?
Your query indicated "The angles should be 45 degrees for both." Does this mean that the two line segments labeled "a" and "b" both form a 45 degree angle with the x axis? If so, that would resolve both of the ambiguities in #'s 4. and 5. above.
After completing the above steps:
Make an attempt to identify the x,y coordinates for both endpoints of the line segment labeled "a". Do the same for the line segment labeled "b".
Using the identified x,y coordinates from the previous step, attempt to calculate the length of the line segment labeled "a". Do the same for the line segment labeled "b".
Edit your query to indicate the resolved ambiguities and show all of your work. In your edited query, indicate very specifically any difficulties that you have.
Leave the last question (i.e. the length of the line segment labeled "c") for last. This seems like a tough one; I'm not sure how to attack it, but others on mathSE will be able to point you to a reference for this problem, or give you a guide on how to attack this last question.
Addendum:
From my perspective, once everything else is resolved, the major obstacle in measuring line "c" will be that you did not include Calculus as one of the tags to this query. Taking this omission at face value, it suggests that your teacher (or your math book) does not intend that Calculus be used.
In order to help MathSE guide you to the intended approach re line "c", it will help if you provide context. That is, what Analytical Geometry theorems have you been taught that you think might be relevant here?