If the normals at x i - y i - where- i -1-2-3-4 on the rectangular hyperbola xy - c 2 meet at - - and x 1 x 2 x 3 x 4- a and y 1 y 2 y 3 y 4- b-then a - b isA- -2-2B- -2-2C- 2 c 4D- -2 c 4

The correct option is D 2c^4The equation of the normal to the hyperbola xy = c^2 at ( ct, ct ) is xt^3 yt ct^4 + c = 0 ⇒ ct^4 xt^3 + yt c = 0 which is ...

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Byju's Answer

Standard XII

Mathematics

Rectangular Hyperbola

If the normal...

Question

If the normals at $(x_{i}, y_{i}), where, i = 1, 2, 3, 4$ on the rectangular hyperbola $x y = c^{2}$ meet at $(α, β) .$ and $x_{1} x_{2} x_{3} x_{4} = a$ and $y_{1} y_{2} y_{3} y_{4} = b$ , then $a + b$ is

α^{2} + β^{2}

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- (α^{2} + β^{2})

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2 c^{4}

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- 2 c^{4}

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Solution

The correct option is D $- 2 c^{4}$
The equation of the normal to the hyperbola $x y = c^{2}$ at $(c t, \frac{c}{t})$ is
$x t^{3} - y t - c t^{4} + c = 0$
$\Rightarrow c t^{4} - x t^{3} + y t - c = 0$
which is passing through $(α, β)$
Thus, $c t^{4} - α t^{3} + β t - c = 0.$
Let its four roots are $t_{1}, t_{2}, t_{3}, t_{4} .$
Therefore, $t_{1} + t_{2} + t_{3} + t_{4} = \frac{α}{c},$
$\sum (t_{1} t_{2}) = 0, \sum (t_{1} t_{2} t_{3}) = - \frac{β}{c}$
and $\sum (t_{1} t_{2} t_{3} t_{4}) = - 1.$
$x_{1} \cdot x_{2} \cdot x_{3} \cdot x_{4} = c^{4} (t_{1} t_{2} t_{3} t_{4}) = - c^{4}$
$y_{1} \cdot y_{2} \cdot y_{3} \cdot y_{4} = c^{4} (\frac{1}{t_{1} t_{2} t_{3} t_{4}}) = c^{4} (\frac{1}{- 1}) = - c^{4}$
$\Rightarrow a + b = - 2 c^{4}$

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If the normals at (xi,yi), where, i=1,2,3,4 on the rectangular hyperbola xy=c2 meet at (α,β). and x1x2x3x4=a and y1y2y3y4=b, then a+b is

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