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Question

The line lx + my + n = 0 will be a normal to the hyperbola b2x2a2y2=a2b2, if

A
a2l2+b2m2=(a2+b2)2n2
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B
a2l2+b2m2=(a2b2)2n2
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C
a2l2+b2m2=(a2+b2)2n
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D
None of these
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Solution

The correct option is D None of these
The equation of the normal at (asecϕ,btanϕ) to the hyperbola x2a2y2b2=1 is
ax sinϕ+by=(a2+b2)tanϕ ......(i)
and the equation of the line is
lx + my + n =0 …… (ii)
now, equations (i) and (ii) represent the same line, therefore
2sinϕl=bm=(a2+b2)tanϕn=(a2+b2)sinϕncosϕ
sinϕ=blamand cosϕ=(a2+b2)lna
a2l2b2m2=(a2+b2)2n2

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