21. -/5 points the equation x² - xy + y² = 7 represents a \rotated ellipse,\ that is, an ellipse whose axes…

21. -/5 points the equation x² - xy + y² = 7 represents a \rotated ellipse,\ that is, an ellipse whose axes are not parallel to the coordinate axes. find the points at which this ellipse crosses the x - axis. smaller x - value (x, y) = ( ) larger x - value (x, y) = ( ) show that the tangent lines at these points are parallel. smaller x - value y = larger x - value y = since these two values are equal, the tangent lines select parallel. resources read it watch it

21. -/5 points the equation x² - xy + y² = 7 represents a \rotated ellipse,\ that is, an ellipse whose axes are not parallel to the coordinate axes. find the points at which this ellipse crosses the x - axis. smaller x - value (x, y) = ( ) larger x - value (x, y) = ( ) show that the tangent lines at these points are parallel. smaller x - value y = larger x - value y = since these two values are equal, the tangent lines select parallel. resources read it watch it

Answer

Explanation:

Step1: Find intersection with x - axis

Set (y = 0) in the equation (x^{2}-xy + y^{2}=7). We get (x^{2}=7), so (x=\pm\sqrt{7}). The points are ((-\sqrt{7},0)) (smaller (x) - value) and ((\sqrt{7},0)) (larger (x) - value).

Step2: Differentiate implicitly

Differentiate (x^{2}-xy + y^{2}=7) with respect to (x). Using the product rule ((uv)^\prime = u^\prime v+uv^\prime) where (u=-x) and (v = y), we have (2x-(y+xy^\prime)+2yy^\prime = 0). Rearrange to solve for (y^\prime): (2x - y-xy^\prime+2yy^\prime = 0), then (y^\prime(2y - x)=y - 2x), so (y^\prime=\frac{y - 2x}{2y - x}).

Step3: Evaluate (y^\prime) at the points

For the point ((-\sqrt{7},0)), substitute (x =-\sqrt{7}) and (y = 0) into (y^\prime): (y^\prime=\frac{0-2(-\sqrt{7})}{2(0)-(-\sqrt{7})}=\frac{2\sqrt{7}}{\sqrt{7}} = 2). For the point ((\sqrt{7},0)), substitute (x=\sqrt{7}) and (y = 0) into (y^\prime): (y^\prime=\frac{0 - 2\sqrt{7}}{2(0)-\sqrt{7}}=\frac{-2\sqrt{7}}{-\sqrt{7}}=2).

Answer:

Smaller (x) - value: ((x,y)=(-\sqrt{7},0)) Larger (x) - value: ((x,y)=(\sqrt{7},0)) Smaller (x) - value (y^\prime = 2) Larger (x) - value (y^\prime = 2) Since these two values are equal, the tangent lines are parallel.