Question

The normal to the curve y(x – 2)(x – 3) = x + 6 at the point where the curve intersects the y-axis passes through the point

$open parentheses fraction numerator 1 over denominator 2 comma end fraction 1 third close parentheses$

$open parentheses fraction numerator 1 over denominator 2 comma end fraction 1 half close parentheses$
$open parentheses negative fraction numerator 1 over denominator 2 comma end fraction minus 3 over 2 close parentheses$

Solution

open parentheses fraction numerator 1 over denominator 2 comma end fraction 1 half close parentheses

space space straight y equals space fraction numerator straight x space plus space 6 over denominator left parenthesis straight x space minus 2 right parenthesis left parenthesis straight x minus 3 right parenthesis end fraction Point space of space intersection space with space straight y space minus space axis space left parenthesis 0 comma 1 right parenthesis straight y apostrophe space equals space fraction numerator left parenthesis straight x squared space minus 5 straight x space plus 6 right parenthesis space left parenthesis 1 right parenthesis space minus space left parenthesis straight x plus space 6 right parenthesis left parenthesis 2 straight x minus 5 right parenthesis over denominator left parenthesis straight x squared minus 5 straight x space plus 6 right parenthesis squared end fraction

y'= 1 at point (0, 1)
∴ Slope of normal is –1
Hence equation of normal is x + y = 1
∴ (1/2, 1/2)satisfy it.

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