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Application of Derivatives

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Short Answer Type

61.

Find the slope of the normal at the point (am³ , am² ) to the curve a x² = y³.

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62.

Find a point on the curve y = (x – 2)² at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4).

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63.
Find the point at which the tangent to the curve $straight y equals space square root of 4 straight x minus 3 end root space minus 1$ has its slope $2 over 3.$

The equation of curve is
$straight y space equals space square root of 4 straight x minus 3 end root space minus space 1 space equals space left parenthesis 4 straight x minus 3 right parenthesis to the power of 1 half end exponent space minus space 1$
$therefore space space space space space space dy over dx space equals space 1 half left parenthesis 4 straight x minus 3 right parenthesis to the power of 1 half end exponent space 4 space minus 0 space equals space fraction numerator 2 over denominator square root of 4 straight x minus 3 end root end fraction.$
which is slope of tangent to the curve.
From given condition.
$dy over dx space equals space 2 over 3 space space space space space space space or space space space space space space space fraction numerator 2 over denominator square root of 4 straight x minus 3 end root end fraction space equals space 2 over 3 space space space space rightwards double arrow space space space space fraction numerator 1 over denominator square root of 4 straight x minus 3 end root end fraction space equals space 1 third rightwards double arrow space space space space space square root of 4 straight x minus 3 end root space equals space 3 space space rightwards double arrow space space space space 4 straight x minus 3 space equals space 9 space space space space space space space rightwards double arrow space space space space 4 straight x space equals space 12 space space space space space space rightwards double arrow space space space space space straight x space equals space 3 When space straight x space equals 3 comma space space space straight y space equals space square root of 4 space cross times 3 minus 3 end root space minus 1 space equals space square root of 9 space minus space 1 space equals space 3 minus 1 space equals space 2 therefore space space space space space required space point space is space left parenthesis 3 comma space 2 right parenthesis.$

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64.

Find the equation of the normal to the curve y² = 4x at the point (1, 2).

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65.

Find the equation of tangent to the ellipse $straight x squared over straight a squared plus straight y squared over straight b squared space equals space 1 space space space at space left parenthesis straight x subscript 1 comma space space straight y subscript 1 right parenthesis.$

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66.

Find the equation of tangent and normal to the hyperbola $straight x squared over straight a squared minus straight y squared over straight b squared space equals space 1 space space at space space left parenthesis straight x subscript 0 comma space space straight y subscript 0 right parenthesis.$

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67. Find the equation of the normal at the point (am², am³) for the curve ay²= x³.

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68.

Find the equation of the tangent and the normal to the curve $straight x to the power of 2 over 3 end exponent plus straight y to the power of 2 over 3 end exponent space equals space 2 space space at space left parenthesis 1 comma space 1 right parenthesis$