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Numerade Educator



Problem 74 Hard Difficulty

(a) Where does the normal line to the ellipse $ x^2 - xy + y^2 = 3 $ at the point (-1, 1) intersect the ellipse a second time?
(b) Illustrate part (a) by graphing the ellipse and the normal line.


(A) $$

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Video Transcript

in this problem were given any question of him. Uh, ellipse. We know that normal line to that. Elves passes Jew this point and were asked, defying a second point that the same normal lead. That's true. So first, take narrative off all to suspect X so that we could find the slope of attention. Lenny had to x minus X months. X and Y Prime was free. Um, well, there's what's great so that it's too white on white crime z zero from this refund very to my front to beat y minus two x Divided by two Wyman's ex less ill Let this tangent line at a given point. So why prime you elect updated to want one. Gotta be one so that the smoke often change in line. Slow engine light. It's one a normal line intention line. Our struggle there for practically each other, meaning that if you work too much spice for book tension with the slope of normal line, we would get negative one. So it means that the slope of normal line should be negative. One that if the slopes of normal line is negative, unless right on equation, any line will be up to form Y minus one is m times X minus X notes. You know the point that that normal and pass sisters are? Why my swan? Now we've found the slope to be negative one times X plus one. From this, we see that the question of the formula is why is a minute thanks. Now, using this whenever VC want in ordinary question, we're just gonna play great extent we can solve for X. So we have sprayed minus extension eight eggs. Expert is Eagle three. Promise. We see that three expert busy three ex crazy with one So that X should be one or X. Should be negative one. Don't forget, we're given a point. And this is one of the points that normal blood passes true. So this point is given and we're interested in this point. If X is one using this line since that is on this line, why should be negative one? So the normal line, um, courses secure first at negative one and one. And that one and negative horn be Julie. We can wrap it. This is the corner and system. This is our lives. You could use any only function Porter. This is the first point. Got this normal life presents, You understand? Second corn, and that would be our normal life.