USING DERIVATIVES FIND EQUATION OF TANGENT AND NORMAL

Problem 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

Problem 2 :

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

Solution

Problem 3 :

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

Solution

Problem 4 :

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

Solution

Problem 5 :

The normal at the point (1, 1) on the curve 2y + x² = 3 is

(a) x + y = 0     (b) x - y = 0    (c) x + y + 1 = 0  (d) x - y = 0

Solution

Problem 6 :

Find the points on the curve

x²/9 + y²/16 = 1

at which the tangents are parallel to Y-axis

Solution

Problem 1 :

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

Problem 2 :

Find the tangent and normal to the curve x³ + y³ = 2xy at (1, 1).

Solution

Problem 3 :

Consider the curve defined by 2y³ + 6x² y - 12x² + 6y = 1

i) Find dy/dx

ii)  Write an equation of each horizontal tangent line to the curve.

iii) The line through the origin with slope -1 is tangent to the curve at point P. Find the x and y coordinate of P.

Solution

Problem 1 :

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

Problem 2 :

Find the equations of the tangents to the curve

y  =  (x+1)/(x-1)

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

Solution

Problem 3 :

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

Problem 4 :

Find the angle between the rectangular hyperbola xy = 2 and the parabola x² + 4y = 0 .

Solution

Problem 5 :

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

Solution

Example 1 :

Find the points on curve

x²-y²  =  2

at which the slope of the tangent is 2.

Solution

Example 2 :

Find at what points on a circle

x²+y²  =  13

the slope of the tangent is -2/3.

Solution

Example 3 :

For the curve y = 4x³ -2x⁵, find all the points at which the tangent passes through the origin.

Solution

Example 4 :

The curve

y = ax³ + bx² + cx + 5

touches the x-axis at the point (-2, 0) and Cuts the y-axis at a point where the slope is 3. Find a, b, c

Solution

Example 5 :

The point on the curve y = x³ – 11x + 5 at which the tangent is y = x –11 is

Solution

Example 6 :

The line y = x + 1 is a tangent to the curve y² = 4x at the point

(a) (1, 2)     (b) (2, 1)    (c) (1, – 2)   (d) (–1 ,2)

Solution

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