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Application of Derivatives

Name: Zigya - For The Curious Learner
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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.

The equation of curve is
 y = ax³ + bx² + cx + 5 ...(1)
It meets y-axis at Q where x = 0
$therefore$ putting x = 0 in (1), we get,
$straight y space equals space 0 plus 0 plus 0 plus 5 space equals space 5$
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$therefore space space space space 12 straight a minus 4 straight b plus 3 space equals space 0$ ...(2)
Also (-2, 0) lies on (1)
$therefore space space 0 space equals space minus 8 straight a plus 4 straight b minus 2 straight c plus 5 space space space space space space space space space space space space space$
or $8 straight a minus 4 straight b plus 1 space equals space 0$
Subtracting (3) from (2), 4 a +2 = 0 $rightwards double arrow space space space space straight a space equals space minus 1 half$
From (2), -6 - 4b + 3 = 0 $rightwards double arrow space space space 4 straight b space equals space minus 3 space space space rightwards double arrow space space space straight b space equals space minus 3 over 4$
$therefore space space space straight a space equals space minus 1 half comma space space space space straight b space equals space minus 3 over 4 comma space space space straight c space equals 3$