Tangent drawn from point c - d to the hyperbola x 2-25- y 2-16-1 make angles - and - with the x - axis- If tan-tan-1- then the value of c 2- d 2

Any tangent to hyperbola is y = mx ±√(m^2 a^2 b^2) Since, its passes through (c,d) ∴ d = mc ±√(a^2 m^2 b^2) ⇒ (d mc)^2 = a^2 m^2 b^2 ⇒ (c^2 a^2) m^2 2c ...

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Standard XII

Mathematics

Slope Form of Tangent: Hyperbola

Tangent drawn...

Question

Tangent drawn from point $(c, d)$ to the hyperbola $\frac{x^{2}}{25} - \frac{y^{2}}{16} = 1$ make angles $α$ and $β$ with the $x -$ axis. If $tan α tan β = 1$ , then the value of $c^{2} - d^{2}$

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Solution

Any tangent to hyperbola is
$y = m x \pm \sqrt{m^{2} a^{2} - b^{2}}$
Since, its passes through $(c, d)$
$∴ d = m c \pm \sqrt{a^{2} m^{2} - b^{2}}$
$\Rightarrow (d - m c)^{2} = a^{2} m^{2} - b^{2}$
$\Rightarrow (c^{2} - a^{2}) m^{2} - 2 c d m + b^{2} + d^{2} = 0$
Product of roots
$m_{1} m_{2} = tan α tan β \Rightarrow \frac{d^{2} + b^{2}}{c^{2} - a^{2}} = 1$
$\Rightarrow d^{2} + b^{2} = c^{2} - a^{2}$
$\Rightarrow c^{2} - d^{2} = a^{2} + b^{2} = 25 + 16 = 41$

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Slope Form of Tangent: Hyperbola

Standard XII Mathematics

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Tangent drawn from point (c,d) to the hyperbola x225−y216=1 make angles α and β with the x− axis. If tanαtanβ=1, then the value of c2−d2

Any tangent to hyperbola is y=mx±√m2a2−b2 Since, its passes through (c,d) ∴d=mc±√a2m2−b2 ⇒(d−mc)2=a2m2−b2 ⇒(c2−a2)m2−2cdm+b2+d2=0 Product of roots m1m2=tanαtanβ⇒d2+b2c2−a2=1 ⇒d2+b2=c2−a2 ⇒c2−d2=a2+b2=25+16=41

Tangent drawn from point $(c, d)$ to the hyperbola $\frac{x^{2}}{25} - \frac{y^{2}}{16} = 1$ make angles $α$ and $β$ with the $x -$ axis. If $tan α tan β = 1$ , then the value of $c^{2} - d^{2}$