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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

The equation of curve is y = x² – 2x + 7 ...(1)
$therefore space space space space space dy over dx space equals space 2 straight x minus 2$
$therefore space space space at space left parenthesis straight x comma space straight y right parenthesis space slope space of space tangent space space equals space 2 straight x minus 2$

(i) consider the line $2 straight x minus straight y plus 9 space equals space 0$ ...(2)
Its slope $equals negative fraction numerator 2 over denominator negative 1 end fraction space equals space 2$
Since tangent to the curve is parallel to line (2)
$therefore space space space space their space slopes space are space space equal space therefore space space space space 2 straight x minus 2 space equals space 2 space space space space space rightwards double arrow space space space space 2 straight x space equals space 4 space space space rightwards double arrow space space space straight x space equals space 2 therefore space space space space from space left parenthesis 1 right parenthesis comma space space space straight y space equals space 4 minus 4 plus 7 space equals space 7 therefore space space space space space space point space is space left parenthesis 2 comma space 7 right parenthesis therefore space space space space space space space equation space of space tangent space at space left parenthesis 2 comma space 7 right parenthesis space with space slope space 2 space is space space space space space space space space space space space space space space space space space space space space straight y minus 7 space equals space 2 left parenthesis straight x minus 2 right parenthesis comma space space space or space space space straight y minus 7 space equals space 2 straight x minus 4 space space space or space space space 2 straight x minus straight y plus 3 space equals space 0$

(ii) Consider the line $5 straight y minus 15 straight x space equals space 13$
$or space space 15 straight x minus 5 straight y plus 13 space equals space 0$ ...(3)
Its slope $equals negative fraction numerator 15 over denominator negative 5 end fraction space equals 3$
Since target to the curve is perpendicular to line (3)
$therefore space space space space space product space of space their space slopes space space equals space minus 1 therefore space space space space space space space space left parenthesis 2 straight x minus 2 right parenthesis thin space left parenthesis 3 right parenthesis space space equals space minus 1 space space space space rightwards double arrow space space 6 straight x minus 6 space equals space minus 1 space space space rightwards double arrow space space 6 straight x space equals space 5 therefore space space space space space straight x space equals space 5 over 6 From space left parenthesis 1 right parenthesis comma space space space straight y space equals space 25 over 36 minus 5 over 3 plus 7 space equals space fraction numerator 45 minus 60 plus 252 over denominator 36 end fraction space equals space 217 over 36 therefore space space space point space is space open parentheses 5 over 6 comma space space 217 over 36 close parentheses therefore space space space space space space equation space of space tangent space at space open parentheses 5 over 6 comma space space 217 over 36 close parentheses space with space slope space minus 1 third space is space space space space space space space space space space space space space space space space space space space space space straight y minus 217 over 36 space equals negative 1 third open parentheses straight x minus 5 over 6 close parentheses space space space or space space space straight y minus 217 over 36 space equals negative straight x over 3 plus 5 over 18 or space space space space 36 straight y minus 217 space equals space minus 12 straight x plus 10 space space space space space space space or space space space space space space 12 straight x plus 36 straight y minus 227 space equals space 0$

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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.

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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.