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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).

The equation of curve is
y² = 4x
Differentiating both sides w.r.t. x, we get,
$2 straight y dy over dx space equals space 4 space space space space space space space or space space space dy over dx space equals space 2 over straight y$
$therefore space space space space at space left parenthesis straight x comma space straight y right parenthesis. space slope space of space tangent space space equals space 2 over straight y therefore space space space space space at space left parenthesis 1 comma space 2 right parenthesis comma space slope space of space tangent space equals space 2 over 2 space equals space 1 therefore space space space space space slope space of space normal space space equals space minus 1 over 1 space equals space minus 1 therefore space space space space space equation space of space normal space at space left parenthesis 1 comma space 2 right parenthesis space is space space space space space space space space space space space space space space space space space space space space space space space space space straight y space minus space 2 space equals space minus space 1 left parenthesis straight x minus 1 right parenthesis space space space space space space or space space space straight y minus 2 space equals space space minus straight x plus 1 or space space space space space space space space space straight x plus straight y minus 3 space equals space 0.$

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