HW 110 Finding the equation of a tangent line to the point that is on given ellipse. We can draw tangent at every point that is on the ellipse. Finding the equation of a tangent at any of the four vertices is simple, it would either be horizontal line, or vertical line. We need to have a good sketch to find the tangent lines: (x-8)2 0+1)2 64 48 - = 1, we want to find an equation of a tangent line to ellipse at point (12,5) This point is on the ellipse, so we can find the equation of a tangent line by the following : (12,5) V1:(0,-1) Inside (12,-1) V2(16,-1) Outside (X, -1) In the above sketch, we have a majoraxis, minoraxis, point (12, 5) which is on the ellipse, we have V1 and V2. When we sketch the tangent line it will intersect the major axis at a point (X, -1) which is located ooutside and on the major axis and to the right of V2. We also need a point (12, -1) , which is located dirrectly under the point (12, 5) and inside the ellipse and on the major axis. We need the diagram and we need 4 points on the major axis or the minor axis, 2 points are the vertices, one of the point lies outside on the major or the minor axis, and one point lies inside and on the major or the minor axis. We need to use all 4 points on the major axis or on the minor axis. In the above diagram we are using the points that are on the major axis. Now to find the cordinates of the out side point on the major axis: since major axis is horizontal, we need to find the following distance and use the following proportion: Horizontal distance between inside point and V1 _Horizontal distance between outside point and V1 Horizontal distance between inside point and V2 Horizontal distance between outside point and V2 That gives us the following ratio:1 = * and when we solve for x we get x = 24. So the outside point is (24, -1), now we have 2 points on the tangent line and we can find the slope of the tangent line and then write an equation of the tangent line: Slope is -, and the equation of the tangent line is y - 5 = 2 (x - 12) ORy + 1 = 2 (x -24) You try the following problems for the HW: For the following equation of the ellipse: (x - 1)2 (y - 10)2 81 45 Find the center, a, b and c value, 4 verticies, 2 foci, area and the ecentricity Sketch the ellipse, labeling major axis, minor axis, and the two verticies on the major axis: 1.) Verify that point (7, 15) is on the ellipse? 2.) Plot the point (7, 15) on the ellipse 3.) Sketch the tangent line and label the two points on the major axis one inside and one outside. (Inside point you know both x and the y value of the point, outside point you only know the y value) 4.) Set up the proportion and solve for x 5.) Find the slope of the tangent line 6.) Find the equation of the tangent line B.) Draw the ellipse again, and repeat the steps 1 to 6 for the point (-5, 5) C.) Draw the ellipse again and repeat the steps 1 to 6 for the point (7, 5) D.) Draw the ellipse again and repeat the steps 1 to 6 for the point (-5, 15) E.) Chalange problem instead of using the major axix use the minor axis to find the tangent line at point (7, 15). You will need to find a vertical distance to V3 and V4, the inside and outside points will be on the minor axis, and when seting the ratio you are finding the vertical distance. (this distance will have radical in it, bur when you solve for the equation it will not have radical in it. (Good luck.)