If P is a point on the hyperbola $$x2-y2=a2$$,C is its centre and S,S' are two foci, then SP⋅S'$$P=$$

Question

If P is a point on the hyperbola $$x2-y2=a2$$,C is its centre and S,S' are two foci, then SP⋅S'$$P=$$

Infinity Learn · Accepted Answer

$$x2-y2=a2$$,$$C(0$$,$$0)e=2Pasec$$θ,atanθ$$s2a$$,$$0s1$$−$$2a$$,$$0SP=asec$$θ−$$a22+atan$$θ$$2SP=a2sec2$$θ$$+2$$−$$22sec$$θ$$+tan2$$θ$$=a1+2tan2$$θ$$+2$$−$$22sec$$θ$$SP=a3+2tan2$$θ−$$22sec$$θS'$$P=(asec$$θ$$+a2)2+(atan$$θ$$)2$$                     $$=sec2$$θ$$+tan2$$θ$$+2+22sec$$θ $$Step-III$$:   $$SPS1P=a29+4tan4$$θ$$+12tan2$$θ$$-8sec2$$θ$$=a24tan4$$θ$$+9+4tan2$$θ−$$8sec2$$θ−$$tan2$$θ$$=a22tan2$$θ$$+12SPS1P=a21+2tan2$$θ……(1) $$Step-IV$$:   $$CP=(asec$$θ$$)2+(atan$$θ$$)2=asec2$$θ$$+tan2$$θ$$=a1+2tan2$$θ$$CP2=a21+2tan2$$θ………(2) From $$1)$$&¯$$(2)SPS1P=(CP)2$$

$If P is a point on the hyperbola x^{2} - y^{2} = a^{2}, C is its centre and S, S^{'} are two foci, then$ $S P \cdot S' P =$

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If P is a point on the hyperbola x2-y2=a2,C is its centre and S,S' are two foci, then SP⋅S'P=

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$If P is a point on the hyperbola x^{2} - y^{2} = a^{2}, C is its centre and S, S^{'} are two foci, then$ $S P \cdot S' P =$