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10-2. Parabola, Ellipse, Hyperbola
normal
The line $y = mx + c$ is a normal to the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{a^2}}} = 1$, if $c = $
A
$ - (2am + b{m^2})$
B
$\frac{{({a^2} + {b^2})m}}{{\sqrt {{a^2} + {b^2}{m^2}} }}$
C
$ - \frac{{({a^2} - {b^2})m}}{{\sqrt {{a^2} + {b^2}{m^2}} }}$
D
$\frac{{({a^2} - {b^2})m}}{{\sqrt {{a^2} + {b^2}} }}$
Solution
(c) As we know that the line $lx + my + n = 0$ is normal to $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$,
if $\frac{{{a^2}}}{{{l^2}}} + \frac{{{b^2}}}{{{m^2}}} = \frac{{{{({a^2} – {b^2})}^2}}}{{{n^2}}}$.
But in this condition, we have to replace $l$ by $m$, $m$ by $-1$ and $n$ by $c$,
then the required condition is $c = \pm \frac{{({a^2} – {b^2})m}}{{\sqrt {{a^2} + {b^2}{m^2}} }}$.
Standard 11
Mathematics
