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Application of Derivatives

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Short Answer Type

121.

Find points on the curve $straight x squared over 4 plus straight y squared over 25 space equals space 1$ at which the tangents are (i) parallel to the x-axis (ii) parallel to the y-axis.

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Long Answer Type

122.

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

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123.

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

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Multiple Choice Questions

124. The points on the curve 9y² = x³, where the normal to the curve make equal intercepts with the axes are

$open parentheses 4 comma space plus-or-minus 8 over 3 close parentheses$
$open parentheses 4 comma space space minus 8 over 3 close parentheses$
$open parentheses 4 comma space plus-or-minus 3 over 8 close parentheses$
$open parentheses 4 comma space plus-or-minus 3 over 8 close parentheses$

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Long Answer Type

125. The curve y = ax³ + bx² + cx + 5 touches the x -axis at P (– 2, 0) and cuts the y-axis at a point Q where its gradient is 3. Find a. b, c.

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126.

Show that the curves 2x = y² and 2xy = k cut at right angles if k² = 8

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127. Prove that the curves x = y² and xy = k cut at right angles if 8k² = 1.

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128.
Show that the curve xy = a² and x² + y² = 2a² touch each other.

The given curves are
$xy space equals straight a squared$ ...(1)
  x² + y² = 2a² ...(2)
Now $left parenthesis straight x plus straight y right parenthesis squared space equals space straight x squared plus straight y squared plus 2 xy space equals space 2 straight a squared plus 2 straight a squared$ $open square brackets because space of space left parenthesis 1 right parenthesis comma space space left parenthesis 2 right parenthesis close square brackets$
$therefore space space space space space space left parenthesis straight x plus straight y right parenthesis squared space equals space 4 straight a squared$
$rightwards double arrow space space space space space straight x plus straight y equals 2 straight a comma space space minus 2 straight a$ ...(3)
Also $left parenthesis straight x minus straight y right parenthesis squared space equals space straight x squared plus straight y squared minus 2 xy$
   $equals 2 straight a squared minus 2 straight a squared space space space space space space space space space space space space space space space space$ $open square brackets because space space space of space left parenthesis 1 right parenthesis comma space left parenthesis 2 right parenthesis close square brackets$
=0
$therefore space space space space space space straight x minus straight y space equals space 0$ ...(4)
Adding (3) and (4), we get,
$2 straight x space equals 2 straight a comma space space space space space space minus 2 straight a space space space space space rightwards double arrow space space space space straight x space equals space straight a comma space space space minus straight a$
$therefore space space space space from space left parenthesis 4 right parenthesis comma space space space space straight y space equals space straight a comma space space space space minus straight a therefore space space space curves space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis space intersect space at space left parenthesis straight a comma space straight a right parenthesis space and space left parenthesis negative straight a comma space minus straight a right parenthesis From space left parenthesis 1 right parenthesis comma space straight y space equals space straight a squared over straight x therefore space space space space dy over dx space equals space minus straight a squared over straight x squared From space left parenthesis 2 right parenthesis comma space space space 2 straight x plus 2 straight y dy over dx space equals space 0 rightwards double arrow space space space space space space dy over dx space equals space minus straight x over straight y Let space space straight m subscript 1 comma space space straight m subscript 2 space be space slopes space of space curves space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis At space left parenthesis straight a comma space straight a right parenthesis space space space space straight m subscript 1 space equals space minus straight a squared over straight a squared equals negative 1 comma space space space space space straight m subscript 2 space equals space minus straight a over straight a space equals space minus 1 therefore space space at space left parenthesis straight a comma space straight a right parenthesis space curves space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis space touch space At space left parenthesis negative straight a comma space minus straight a right parenthesis space space space space straight m subscript 1 space equals space minus straight a squared over straight a squared space equals space minus 1 comma space space space straight m subscript 2 space equals space minus fraction numerator negative straight a over denominator negative straight a end fraction equals negative 1 therefore space space space at space left parenthesis negative straight a comma space space space minus straight a right parenthesis comma space curves space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis space touch.$