USING DERIVATIVES FIND EQUATION OF TANGENT AND NORMAL

(1)  Find the slope of the tangent to the curves at the respective given points.

(i) y = x⁴ + 2x² − x at x = 1

(ii)  x  =  a cos³t, y = b sin³t at t  =  π/2

Solution

(2)  Find the point on the curve y = x² − 5x + 4 at which the tangent is parallel to the line 3x + y = 7 .

Solution

(3)  Find the points on the curve y = x³ − 6x² + x + 3 where the normal is parallel to the line x + y  =  1729.

Solution

(4)  Find the points on the curve y² - 4xy  =  x²+ 5 for which the tangent is horizontal.

Solution

(5)  Find the tangent and normal to the following curves at the given points on the curve.

(i)  y  =  x² - x⁴ at (1, 0)

(ii)  y  =  x⁴+2e^x at (0, 2)

(iii)  y  =  x sin x at (π/2, π/2)

(iv)  x  =  cost, y  =  2 sin²t at t  =  π/3

Solution

(6)  Find the equations of the tangents to the curve

y  =  1+x³

for which the tangent is orthogonal with the line

x+12y  =  12

Solution

(7)  Find the equations of the tangents to the curve

y  =  (x+1)/(x-1)

which are parallel to the line x+2y  =  6.

Solution

(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 x² + 4y = 0 .

Solution

(10)  Show that the two curves x² − y² = r² and xy = c² where c, r are constants, cut orthogonally 

Solution

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