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Tangents drawn from the point (c, d) to the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1` make angles `alpha` and `beta` with the x-axis.
If `tan alpha tan beta=1`, then find the value of `c^(2)-d^(2)`.

Text Solution

Verified by Experts

One of the equations of tangents to the hyperbola having slope m is `y=mx+sqrt(a^(2)m^(2)-b^(2))`. It passes through (c, d). So,
`d=mc+sqrt(a^(2)m^(2)-b^(2))`
`"or "(d-mc)^(2)=a^(2)m^(2)-b^(2)`
`"or "(c^(2)-a^(2))m^(2)-2cdm+d^(2)+b^(2)=0`
`"or Product of roots"=m_(1)m_(2)=(a^(2)+b^(2))/(c^(2))-a^(2)`
`"or "tan alpha tan beta=(d^(2)+b^(2))/(c^(2)-a^(2))=1`
`"or "d^(2)+b^(2)=c^(2)-a^(2)`
`"or "c^(2)-d^(2)=a^(2)+b^(2)`
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