Answer:
This is the equation of the normal line at the point (2, 1) on the ellipse (x^2)/4 + y^2 = 2.
Explanation:
The equation of the ellipse is (x^2)/4 + y^2 = 2. To find the equation of the normal line at the point (2, 1), we first need to find the slope of the tangent line at that point. To do this, we can use the formula for the slope of a tangent line:
slope = -(d/dx)(f(x)) / (d/dy)(f(y))
Where f(x) and f(y) are the equations of the ellipse.
So,
slope = -(x/2)/y = -x/2y
Now we can substitute the point (2, 1) into the equation for the slope:
slope = -(2/2)/1 = -1
We know that the slope of the normal line is the negative reciprocal of the slope of the tangent line, so the slope of the normal line is 1.
To find the equation of the normal line, we can use the point-slope form of a line:
y - y1 = m(x - x1)
Where (x1, y1) is the point on the line and m is the slope.
So,
y - 1 = 1(x - 2)
Simplifying, we get:
y = x - 1
This is the equation of the normal line at the point (2, 1) on the ellipse (x^2)/4 + y^2 = 2.