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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.

The equations of curve is $straight y equals straight x cubed minus 3 straight x$ ...(1)
Differentiating w.r.t.x, $dy over dx space equals 3 straight x squared minus 3$
$therefore space space space space at space left parenthesis straight x comma space space straight y right parenthesis comma space space slope space of space tangent space space equals space 3 straight x squared minus 3 space space space rightwards double arrow space space space space space at space left parenthesis straight x comma space straight y right parenthesis comma space slope space of space normal space equals space minus fraction numerator 1 over denominator 3 straight x squared minus 3 end fraction therefore space space space space space space straight m subscript 1 space equals space minus fraction numerator 1 over denominator 3 straight x squared minus 3 end fraction Let space space straight m subscript 2 space be space slope space of space line space straight x plus 9 straight y space equals space 14 therefore space space space space space space straight m subscript 2 space equals space minus 1 over 9 therefore space space space space space space space space normal space is space parallel space to space left parenthesis 2 right parenthesis space space space space space therefore space space space straight m subscript 1 space equals space straight m subscript 2 therefore space space space space space space fraction numerator 1 over denominator 3 straight x squared minus 3 end fraction space equals space minus 1 over 9 space space space space space rightwards double arrow space space space space space space fraction numerator 1 over denominator straight x squared minus 1 end fraction space equals space 1 third rightwards double arrow space space space space space space straight x squared minus 1 space equals space 3 comma space space space space space rightwards double arrow space space space straight x squared space equals space 4 comma space space space space rightwards double arrow space space space space straight x space equals space 2 comma space space minus 2 When space straight x space equals 2 comma space space from space left parenthesis 1 right parenthesis comma space space straight y space equals space 8 minus 6 space equals space 2 therefore space space space points space is space left parenthesis 2 comma space space 2 right parenthesis therefore space space space space equation space of space normal space at space left parenthesis 2 comma space 2 right parenthesis space is space straight y minus 2 space equals space minus 1 over 9 left parenthesis straight x minus 2 right parenthesis comma space space space or space space 9 straight y minus 18 space equals space minus straight x plus 2 or space space space space straight x plus 9 straight y minus 20 space equals space 0 When space straight x space equals space minus 2 comma space from space left parenthesis 1 right parenthesis comma space straight y space equals negative 8 plus 6 space equals space minus 2 therefore space space space space space points space is space left parenthesis negative 2 comma space space space minus 2 right parenthesis therefore space space space space equation space of space normal space at space left parenthesis negative 2 comma space minus 2 right parenthesis space is space straight y minus left parenthesis negative 2 right parenthesis space equals space minus 1 over 9 left square bracket straight x minus left parenthesis negative 2 right parenthesis right square bracket or space space space space space space 9 straight y plus 18 space equals straight x minus 2 space space space space or space space straight x plus 9 straight y plus 20 space equals space 0.$

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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.

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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.

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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.