Zadanie 1b
Find angle BVC.
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Find angle BVC.
Write an expression for (g∘f)(x).
Given (g∘f)(1) = 25, find all possible k.
Find the length BV.
Find angle BVC.
(i) Express the slope of y=kx² using only x,y. (ii) Hence show its orthogonal trajectories satisfy y′=−x/(2y). (iii) Solve this differential equation to show that these trajectories have equation x²+2y²=C, with constant C.
For a=4 and b=√5, sketch x2/25 + y2/16 = 1 and x2/4 − y2/5 = 1 on common axes. Label each curve and its x-intercepts. Intersection coordinates are not required.
Solve E_a and H_b simultaneously to show that their four intersections have coordinates , where the two signs are independent.
Show that E_a and H_b intersect orthogonally at their first-quadrant intersection M. You may use xM2 = (a2 + 9)(9 − b2)/9 and yM2 = a2b2/9.
Now take f=−2x/y and α=π/4, working locally where x>0, y≠0 and y+2x≠0. (i) Show that g=(y−2x)/(y+2x). (ii) Solve the homogeneous equation y′=(y−2x)/(y+2x) and give its general implicit form h(x,y)=d. You may use…