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51.
Find the points on the curve y = x³ at which the slope of the tangent is equal to the y-coordinate of the point.

The equation of curve is $straight y space equals space straight x cubed$
$therefore space space space space dy over dx space equals space 3 straight x squared$
$Let space space left parenthesis straight x subscript 1 comma space space straight y subscript 1 right parenthesis space be space the space point.$
$At space left parenthesis straight x subscript 1 comma space space straight y subscript 1 right parenthesis space dy over dx space equals space 3 straight x subscript 1 squared comma space space space which space is space straight a space slope space of space tangent. space$
From given condition,
$3 straight x subscript 1 squared space equals space straight y subscript 1$ ...(1)
Also, $left parenthesis straight x subscript 1 comma space straight y subscript 1 right parenthesis comma space lies space on space straight y space equals space straight x cubed$
$therefore space space space space space space space straight y subscript 1 space equals space straight x subscript 1 cubed$
From (1) and (2), $straight x subscript 1 cubed space equals space 3 straight x subscript 1 squared space space space space rightwards double arrow space space space space straight x subscript 1 squared left parenthesis straight x subscript 1 minus 3 right parenthesis space equals space 0 space space space space rightwards double arrow space space space space straight x subscript 1 space equals space 0 comma space space 3$
From (1), $straight y subscript 1 space equals 0 comma space 27$
$therefore space space space space space space points space are space left parenthesis 0 comma space 0 right parenthesis comma space space space left parenthesis 3 comma space 27 right parenthesis.$

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52. Find the slope of the tangent to the curve x = at², y = 2at at t = 1.

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53.

Find the slope of the tangent to the curve y = x³ – x at x = 2.

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54.

Find the slope of the tangent to the curve y = 3x⁴ – 4x at (i) x = 1 (ii) x = 4.

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55.

Find the slope of the tangent to the curve $straight y space equals fraction numerator straight x minus 1 over denominator straight x minus 2 end fraction comma space space space straight x not equal to space 2 space space at space straight x space equals space 10$ .

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56. For the curve y = 3x² + 4 x , find the slope of the tangent to the curve at the point whose x-coordrnate is – 2.

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57. Find the slope of the tangent to the curve y = x³ – x + 1 at the point whose x-coordmate is 2.

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58.

Find the slope of the tangent to the curve y = x³ – 3x + 2 at the point whose. x-coordinate is 3.

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59.

Find the slope of the normal to the curve $straight x equals space straight a space cos cubed straight theta comma space space space straight y space equals space straight a space sin cubed straight theta space space space at space straight theta space equals space straight pi over 4$ .

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60.

Find the slope of the normal to the curve $straight x space equals space 1 minus straight a space sin space straight theta comma space space space straight y space equals space space straight b space cos squared straight theta space space at space straight theta space equals space straight pi over 2.$

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51. Find the points on the curve y = x3 at which the slope of the tangent is equal to the y-coordinate of the point.

51.
Find the points on the curve y = x³ at which the slope of the tangent is equal to the y-coordinate of the point.