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Application of Derivatives

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Long Answer Type

111.

Find the equation of the tangent line to the curve y = x² – 2x + 7 which is
(a) parallel to the line 2x – y + 9 = 0
(ii) perpendicular to the line 5y – 15y = 13

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112. Find the equations of normal lines to the curve y = x² – 3 x which are parallel to the line x + 9y = 14.

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113. Find the equation of the normals to the curve y = x³ + 2x + 6 which are parallel to the line 14y + 4 = 0.

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114.

At what point will the tangent to the curve y = 2x³ – 15x² + 36x – 21 be parallel to x-axis ? Also, find the equations of tangents to the curve at those points.

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Short Answer Type

115.

Find the points on the curve x² + y² -2x – 3 = 0 at which the tangents are parallel to the x-axis.

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116. Find points at which the tangent to the curve y = x³ – 3x² – 9x + 7 is parallel to the x-axis.

The equation of curve is
$straight y equals straight x cubed minus 3 straight x squared minus 9 straight x plus 7$
$therefore space space space space space space space space dy over dx space equals space 3 straight x squared minus 6 straight x minus 9$
For the points on the curve, where tangents are parallel to x-axis
$dy over dx space equals space 0 space space space space space rightwards double arrow space space space space 3 straight x squared minus 6 straight x minus 9 space equals space 0 space space space space space rightwards double arrow space space space straight x squared minus 2 straight x minus 3 space equals space 0$

$therefore space space space space left parenthesis straight x minus 3 right parenthesis thin space left parenthesis straight x plus 1 right parenthesis space equals space 0 space space space space space space rightwards double arrow space space space space straight x space equals space 3 comma space space minus 1$
When $straight x space equals 3 comma space space space straight y equals left parenthesis 3 right parenthesis cubed minus 3 left parenthesis 3 right parenthesis squared space minus space 9 left parenthesis 3 right parenthesis plus space 7 space equals space 27 minus 27 minus 27 plus 7 space equals space minus 20$
When $straight x equals negative 1 comma space space straight y space equals space left parenthesis negative 1 right parenthesis cubed minus 3 left parenthesis negative 1 right parenthesis squared minus 9 left parenthesis negative 1 right parenthesis plus 7 space equals negative 1 minus 3 plus 9 plus 7 space equals space 12$
$therefore space space space space space space required space points space are space left parenthesis 3 comma space minus 20 right parenthesis comma space left parenthesis negative 1 comma space 12 right parenthesis.$

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117.

Find point on the curve y = 3x² – 12x + 6 at which tangent is parallel to x-axis. Also find the equation of tangent line.

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118.

Find out on the curve y = x² – x – 8 at which tangent is parallel to x-axis. Also find the equation of tangent line.

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119.

Find out on the curve y = 2 x² – 6x – 4 at which tangent is parallel to x-axis. Also find the equation of tangent line.

74 Views

Long Answer Type

120.

Find out the points on the curve $straight x squared over 9 minus straight y squared over 16 space equals space 1$ at which the tangents are (i) parallel to the x-axis and (ii) parallel to the y-axis.