If the equation of the pair of straight lines passing through the point (1$$,$$1) one making an angle θ with the positive direction of the $$x-axis$$ and the other making the same angle with the positive direction of the $$y-axis$$, is $$x2$$−$$(a+2)xy+y2+a(x+y$$−$$1)=0$$,a≠−$$2$$,then the value of $$sin2$$θ is

Question

If the equation of the pair of straight lines passing through the point (1$$,$$1) one making an angle θ with the positive direction of the $$x-axis$$ and the other making the same angle with the positive direction of the $$y-axis$$, is $$x2$$−$$(a+2)xy+y2+a(x+y$$−$$1)=0$$,a≠−$$2$$,then the value of $$sin2$$θ is

Infinity Learn · Accepted Answer

The equations of the given lines are y−$$1=$$$$tan$$ θ (x$$−$$1) and y−$$1=$$$$cot$$ θ (x$$−$$1)So$$, their joint equation is [$$(y$$−$$1)−$$tan$$ θ (x$$−$$1)][y−$$1$$−$$cot$$ θ (x$$−$$1)]$$=0$$ or (y$$−$$1)2$$−$$($$$$tan$$θ$$+$$$$cot$$ θ$$) $$(x$$−$$1)(y$$−$$1)+(x$$−$$1)2=0or$$ $$x2$$−$$($$$$tan$$θ$$+$$$$cot$$ θ$$)xy+y2+($$$$tan$$ θ$$+$$$$cot$$ θ−$$2)(x+y$$−$$1)=0Comparing$$ with the given equation, we get $$tan$$θ$$+$$$$cot$$ θ$$=a+2$$ or $$1sin$$ θ $$cos$$θ$$=a+2$$ or $$sin$$ $$2$$θ $$=2a+2$$

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If the equation of the pair of straight lines passing through the point (1,1) one making an angle θ with the positive direction of the x-axis and the other making the same angle with the positive direction of the y-axis, is x2−(a+2)xy+y2+a(x+y−1)=0,a≠−2,then the value of sin2θ is

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