$$Statement-1$$: Two tangents drawn from any point on he hyperbola $$x2$$−$$y2=a2$$−$$b2$$ to the ellipse $$x2a2+y2b2$$ $$=1$$ make complementary angles with the axis of the ellipse.$$Statement-2$$: If two lines make complementary angles with the axis of x then the product of their slopes is $$1$$.

Question

$$Statement-1$$: Two tangents drawn from any point on he hyperbola $$x2$$−$$y2=a2$$−$$b2$$ to the ellipse $$x2a2+y2b2$$ $$=1$$ make complementary angles with the axis of the ellipse.$$Statement-2$$: If two lines make complementary angles with the axis of x then the product of their slopes is $$1$$.

Infinity Learn · Accepted Answer

If the angle in $$statement-2$$ are α and β such that α$$+$$β$$=$$π$$/2$$ then the product of the slopes is $$tan$$ α$$tan$$⁡β $$=$$ $$1$$, and the $$statement-2$$ is true.In $$statement-1$$, any tangent to the ellipse $$isy=mx+a2m2+b2$$ which passes $$througha2$$−$$b2sec$$⁡θ,$$a2$$−$$b2tan$$⁡θ⇒$$a2$$−$$b2($$$$tan$$⁡θ−msec⁡θ$$)2=a2m2+b2Product$$ of the slopes $$=a2$$−$$b2tan2$$⁡θ−$$b2a2$$−$$b2sec2$$⁡θ−$$a2=a2sin2$$⁡θ−$$b2a2sin2$$⁡θ−$$b2=1So$$ by $$statement-2$$, $$statement-1$$ is also true.

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Statement-1: Two tangents drawn from any point on he hyperbola x2−y2=a2−b2 to the ellipse x2a2+y2b2 =1 make complementary angles with the axis of the ellipse.Statement-2: If two lines make complementary angles with the axis of x then the product of their slopes is 1.

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