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10-2. Parabola, Ellipse, Hyperbola
medium
The equation of tangent and normal at point $(3, -2)$ of ellipse $4{x^2} + 9{y^2} = 36$ are
A
$\frac{x}{3} - \frac{y}{2} = 1,\;\frac{x}{2} + \frac{y}{3} = \frac{5}{6}$
B
$\frac{x}{3} + \frac{y}{2} = 1,\;\frac{x}{2} - \frac{y}{3} = \frac{5}{6}$
C
$\frac{x}{2} + \frac{y}{3} = 1,\;\frac{x}{3} - \frac{y}{2} = \frac{5}{6}$
D
None of these
Solution
(a) Given, equation of ellipse is $4{x^2} + 9{y^2} = 36$
Tangent at point $(3,-2)$ is $\frac{{(3)x}}{9} + \frac{{( – 2)y}}{4} = 1$ or $\frac{x}{3} – \frac{y}{2} = 1$
$\therefore $Normal is $\frac{x}{2} + \frac{y}{3} = k$ and it passes through point $(3,-2)$
$\therefore $$\frac{3}{2} – \frac{2}{3} = k $
$\Rightarrow k = \frac{5}{6}$
$\therefore $Normal is, $\frac{x}{2} + \frac{y}{3} = \frac{5}{6}$.
Standard 11
Mathematics
