(20 pts) Let the points A(5,1), B(4,-2), and C(-1,3) be points that all lie on the ellipse $x^2 + 3y^2 = 28$. Algebraically find a point D on the ellipse such that CD is parallel to AB.
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First, let's find the equation of the line AB. We can use the slope-intercept form of a line, y = mx + b, where m is the slope and b is the y-intercept. The slope of AB can be found using the formula: m = (y2 - y1) / (x2 - x1), where (x1, y1) = (5, 1) and (x2, Show more…
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(a) Verify that the points $A(5,1), B(4,-2),$ and $C(-1,3)$ all lie on the ellipse $x^{2}+3 y^{2}=28$ (b) Find a point $D$ on the ellipse such that $\overline{C D}$ is parallel to $\overline{A B}$ (c) If $O$ denotes the center of the ellipse, show that the triangles $O A C$ and $O B D$ have equal areas. Suggestion: In computing the areas, the formula given at the end of Exercise 34 in Section 1.4 is useful.
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