Question

1. -/5 points the equation x² - xy + y² = 7 represents a

otated 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

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.