If the pair of lines $$a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$$ intersect on the $$y$$-axis then :
JEE · Math · previous-year question
- A.$$2fgh = b{g^2} + c{h^2}$$correct
- B.$$b{g^2} \ne c{h^2}$$
- C.$$abc = 2fgh$$
- D.none of these
Answer
A. $$2fgh = b{g^2} + c{h^2}$$
Explanation
Put $$x=0$$ in the given equation $$ \Rightarrow b{y^2} + 2fy + c = 0.$$ For unique point of intersection $${f^2} - bc = 0$$ $$ \Rightarrow a{f^2} - abc = 0.$$ Since $$abc + 2fgh - a{f^2} - b{g^2} - c{h^2} = 0$$ $$ \Rightarrow 2fgh - b{g^2} - c{h^2} = 0$$
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