4. (a) Suppose a pair of axes x and y is rotated by a to obtain a new pair of axes, X and Y. Show that x = X cosa - Y sin a and y = X sina + Y cos a. (b) Let C be the conic section with equation 3x² - 2√3xy + y² + 16x + 16√3y = 0. (i) Find a value of a € [0, π) such that the coefficient of XY in the equation for C in X and Y coordinates (as in part (a)) is zero. Find this equation. (ii) Sketch C in the x and y coordinates. Find its eccentricity and (where these exist) its foci, axis-intercepts, and asymptotes.

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