Final answer:
To find the equations for the tangent and normal lines to the curve cos y = √2 x cosx, we need to find the derivative of the curve and determine the slopes. The equation for the tangent line is dy/dx = -√2, and the equation for the normal line is y - y₁ = (1/√2)(x - x₁).
Step-by-step explanation:
Equation for the line tangent to the curve:
To find the equation for the line that is tangent to the curve cos y = √2 x cosx, we need to find the derivative of the given curve. Taking the derivative of both sides with respect to x, we get -sin y * dy/dx = √2 * -sin x. Now, substitute the values of cos y and sin x from the given equation, which gives us -sin y * dy/dx = √2 * sin y. Simplifying further, we get dy/dx = -√2
Equation for the line normal to the curve:
The slope of the line normal to a curve is the negative reciprocal of the slope of the tangent line at that point. Thus, the slope of the normal line is 1/√2. Using the point-slope form of a line, we can write the equation of the line normal to the curve as y - y₁ = m(x - x₁), where (x₁, y₁) is the point of tangency. Substitute the values of x₁, y₁, and m, we get the equation y - y₁ = (1/√2)(x - x₁).