Question

Normal to the curve $x^2=4y$ which passes through the point $(1,2)$

• A. $x+y=3$
• B. $x-y=3$
• C. $2x+y=4$
• D. $x+2y=5$

Answer 1

The correct option is A $x+y=3$

From the equation of the curve $x^2=4y$,
we get
$dy/dx=x/2.$ If $m$ be the slope of the normal to $x^2=4y$
then $m=\frac{-1}{\left(\frac{dy}{dx}\right)}=-\frac{2}{x}$

$\therefore x=\frac{-2}{m}$ and $y=\frac{x^2}{4}=1/m^2$
Thus normal is
$y-\frac{1}{m^2}=m(x+\frac{2}{m})$.

If it passes through the point (1,2),
then $\therefore 2-\frac{1}{m^2}=m(1+\frac{2}{m})=m+2\therefore m^3=-1$

or$m=-1$

$\therefore$Required normal is,
$(y-2)=-1(x-1)$

$\Rightarrow x+y=3$