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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.

73 Views

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

74 Views

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.

The equation of curve is
y = 2x³ – 15x² + 36x – 21
Differentiating both sides w.r.t.x, we get,
$dy over dx space equals space 6 straight x squared minus 30 straight x plus 36$
For the points on the curve, where tangents are parallel to x-axis
$dy over dx space equals space 0 space space space space space space space space space space space space space space space space space space space rightwards double arrow space space space space 6 straight x squared minus 30 straight x plus 36 space equals space 0 space space space rightwards double arrow space space space space straight x squared minus 5 straight x plus 6 space equals space 0$
$therefore space space space space space space left parenthesis straight x minus 2 right parenthesis thin space left parenthesis straight x minus 3 right parenthesis space equals space 0 space space space space space space space space space rightwards double arrow space space space straight x space equals space 2 comma space space 3 therefore space space space space space from space left parenthesis 1 right parenthesis comma space space space straight y space equals space 2 left parenthesis 2 right parenthesis cubed space minus 15 space left parenthesis 2 right parenthesis squared space plus space 36 left parenthesis 2 right parenthesis space minus 21 comma space space space 2 left parenthesis 3 right parenthesis squared space minus space 15 left parenthesis 3 right parenthesis squared space plus 36 left parenthesis 3 right parenthesis minus 21 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 16 minus 60 plus 72 minus 21 comma space space space space 54 minus 135 plus 108 minus 21 space equals space 7 comma space 6 therefore space space space space space space points space are space left parenthesis 2 comma space 7 right parenthesis comma space left parenthesis 3 comma space 6 right parenthesis The space equation space of space tangent space at space left parenthesis 2 comma space 7 right parenthesis space parallel space to space straight x minus axis space is space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space straight y minus 7 space equals space 0 space left parenthesis straight x minus 2 right parenthesis space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space space space straight m space equals space 0 close square brackets or space space space space space space space space space space space space space space space space space space space space space space space space space space space space space straight y minus 7 space equals space 0 The space equation space of space tangent space at space left parenthesis 3 comma space 6 right parenthesis space parallel space to space straight x minus axis space is space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space straight y minus 6 space equals space 0 left parenthesis straight x minus 3 right parenthesis space space space space or space space space straight y minus 6 space equals space 0$

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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.

72 Views

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

93 Views

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.

74 Views

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.

81 Views

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.