## Find the equation of the normals to the curve y = x³ + 2x + 6 which are parallel to the line x + 14y + 4 = 0.

Question

Find the equation of the normals to the curve y = x³ + 2x + 6 which are parallel to the line x + 14y + 4 = 0.

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3 weeks 2021-10-04T01:05:35+00:00 2 Answers 0 views 0

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Step-by-step explanation:

ans:

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2. Equation of the curve is y = x3 + 2x + 6

Slope of the normal at point (x,y) = minus fraction numerator 1 over denominator open parentheses begin display style dy over dx end style close parentheses end fraction

On substitution, we get

Slope of the normal = minus fraction numerator 1 over denominator 3 straight x squared plus 2 end fraction space space space space space….. left parenthesis 1 right parenthesis

Normal to the curve is parallel to the line x + 14y + 4 = 0.

i.e yequals minus 1 over 14 equals minus fraction numerator 1 over denominator 3 straight x squared plus 2 end fraction

So the slope of the line is the slope of the normal.

Slope of the line is minus 1 over 14 equals minus fraction numerator 1 over denominator 3 straight x squared plus 2 end fraction

When x = 2, y = 18 and when x = -2, y =-6

Therefore, there are two normal to the curve y = x3 + 2x + 6.

Equation of normal through point (2,18) is given by:

Therefore, the equation of normal to the curve are x+14y-254 = 0

and x + 14y + 86 = 0