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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 $2 over 3.$

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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 $straight x to the power of 2 over 3 end exponent plus straight y to the power of 2 over 3 end exponent space equals space 2 space space at space left parenthesis 1 comma space 1 right parenthesis$

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69.
Find the equation of the tangent and normal to the curve y² = 4ax at the point (at², 2at).

The equation of curve is y² = 4ax
$therefore space space space space space 2 straight y dy over dx space equals 4 straight a space space space space space rightwards double arrow space space space space space space space dy over dx space equals space fraction numerator 2 straight a over denominator straight y end fraction$
$At space left parenthesis at squared comma space space 2 at right parenthesis comma space space space dy over dx equals fraction numerator 2 straight a over denominator 2 at end fraction space equals space 1 over straight t therefore space space space space space space at space left parenthesis at squared comma space space 2 at right parenthesis comma space space slope space of space tangent space equals space 1 over straight t therefore space space space space equation space of space tangent space to space the space curve space at space left parenthesis at squared comma space space space 2 at right parenthesis space is space space space space space space space space space space space space space space straight y minus 2 at space equals space 1 over straight t left parenthesis straight x minus at squared right parenthesis space space space or space space space space space ty minus 2 at squared space equals space straight x minus at squared space space or space space space ty space equals space straight x plus at squared At space left parenthesis at squared comma space space 2 at right parenthesis comma space slope space of space normal space equals space minus straight t therefore space space space equation space of space normal space to space the space curve space at space left parenthesis at squared comma space space space 2 at right parenthesis space is space space space space space space space space space space space space space space space space space space space space space space space space straight y minus 2 at space equals space minus straight t left parenthesis straight x minus at squared right parenthesis. or space space space space space space space space straight y minus 2 at space equals space minus tx plus at cubed or space space space space space space space space straight y space equals space minus tx plus 2 at plus at cubed.$