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10-2. Parabola, Ellipse, Hyperbola
normal
Extremities of the latera recta of the ellipses $\frac{{{x^2}}}{{{a^2}}}\,\, + \,\,\frac{{{y^2}}}{{{b^2}}}\, = \,1\,$ $(a > b)$ having a given major axis $2a$ lies on
A
$x^2 = a(a - y)$
B
$x^2 = a (a + y)$
C
$y^2 = a(a + x)$
D
both $(A)$ and $(B)$
Solution
$h = \pm ae ; k = \pm \frac{{{b^2}}}{a}\,$
$k = \pm a(1 -e^2) = \pm a \left( {1 – \frac{{{h^2}}}{{{a^2}}}} \right)\,$ $=$ $ \pm$ $\left( {a – \frac{{{h^2}}}{a}} \right)\,$
$+ ve\ sign , k =$ $a – \frac{{{h^2}}}{a}\,\,$ $\Rightarrow$ $\frac{{{h^2}}}{a}\,\, = \,a – k$ $\Rightarrow$ $h^2 = a ( a – k)$ $\Rightarrow$ $(A)$
$- ve\ sign , k = – a + \frac{{{h^2}}}{a}\,\,$ $\Rightarrow$ $h^2 = a (a + k)$
Standard 11
Mathematics
