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Prove that any hyperbola and its conjuga...

Prove that any hyperbola and its conjugate hyperbola cannot have common normal.

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Consider hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1.`
Equation of normal to hyperbola at point `P(a sec theta, b tan theta)` is
`ax cos theta+by cot theta=a^(2)+b^(2)" (1)"`
Equation of normal to hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=-1` at point `Q( a tan phi, b sec phi)` is
`ax cot phi+"by" cos phi=a^(2)+b^(2)" (2)"`
If Eqs. (1) and (2) represent the same straight line, then
`(cot phi)/(cos theta)=(cos phi)/(cot theta)=1`
`rArr" "tan phi = sec theta and sec phi = tan theta`
`rArr" "sec^(2)phi-tan^(2)phi=tan^(2)theta-sec^(2)theta=-1,` which is not possible.
Thus, hyperbola and its conjugate hyperbola cannot have common normal.
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