(1) Find the slope of the tangent to the curves at the respective given points.
(i) y = x^{4} + 2x^{2} − x at x = 1
(ii) x = a cos^{3}t, y = b sin^{3}t at t = π/2
(2) Find the point on the curve y = x^{2} − 5x + 4 at which the tangent is parallel to the line 3x + y = 7 .
(3) Find the points on the curve y = x^{3} − 6x^{2} + x + 3 where the normal is parallel to the line x + y = 1729.
(4) Find the points on the curve y^{2} - 4xy = x^{2}+ 5 for which the tangent is horizontal.
(5) Find the tangent and normal to the following curves at the given points on the curve.
(i) y = x^{2} - x^{4} at (1, 0)
(ii) y = x^{4}+2e^{x} at (0, 2)
(iii) y = x sin x at (π/2, π/2)
(iv) x = cost, y = 2 sin^{2}t at t = π/3
(6) Find the equations of the tangents to the curve
y = 1+x^{3}
for which the tangent is orthogonal with the line
x+12y = 12
(7) Find the equations of the tangents to the curve
y = (x+1)/(x-1)
which are parallel to the line x+2y = 6.
(8) Find the equation of tangent and normal to the curve given by
x = 7 cos t and y = 2 sin t for all t
at any point on the curve. Solution
(9) Find the angle between the rectangular hyperbola xy = 2 and the parabola x^{2} + 4y = 0 .
(10) Show that the two curves x^{2} − y^{2} = r^{2} and xy = c^{2} where c, r are constants, cut orthogonally
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