14. If the equation y = x+a/1+, reg...
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14. If the equation y = x+a/1+, regarded as a quadratic in 2, will have equal roots, then x2 + y2 is equal to (1) -22 (2) a (3) 0

JEE/Engineering Exams
Maths
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[ begin{array}{l}(y-lambda x)=a sqrt{1+lambda^{2}} text { Squaring } qquad begin{array}{l}(y-lambda x)^{2}=a^{2}left(1+lambda^{2}right) y^{2}+lambda^{2} x^{2}-2 lambda x y=a^{2}+a^{2} d^{2}end{array} Rightarrow quad lambda^{2}left(x^{2}-a^{2}right)-2 lambda x y+left(y^{2}-a^{2}right) frac{operatorname{tep}}{2} text { Equal roots means } D=0 4 x^{2} y^{2}-4left(y^{2}-a^{2}right)left(x^{2}-a^{2}right)=0 Rightarrow 4 x^{2} y^{2}-4left[y^{2} x^{2}-a^{2} y^{2}-a^{2} x^{2}+a^{4}right]=0 Rightarrow 4 x^{2} y^{2}-4 x^{2} y^{2}+4 a^{2} y^{2}+4 a^{2} x^{2}-4 a^{4}=0 3 x^{2}+y^{2}=a^{2}end{array}=0 ]
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