question_answer

If the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\]cuts off a chord of length 2b from the line \[y=mx+c\], then 

A)            \[(1-{{m}^{2}})({{a}^{2}}+{{b}^{2}})={{c}^{2}}\]             

B)            \[(1+{{m}^{2}})({{a}^{2}}-{{b}^{2}})={{c}^{2}}\]

C)            \[(1-{{m}^{2}})({{a}^{2}}-{{b}^{2}})={{c}^{2}}\] 

D)            None of these

Correct Answer: B

Solution :

           We know \[CD=\left| \frac{c}{\sqrt{1+{{m}^{2}}}} \right|\]              ?.(i)                    But according to figure,            \[{{a}^{2}}-{{b}^{2}}=C{{D}^{2}}\]            From (i) and (ii), we get \[{{a}^{2}}-{{b}^{2}}=\frac{{{c}^{2}}}{(1+{{m}^{2}})}\]                    \[\Rightarrow ({{a}^{2}}-{{b}^{2}})(1+{{m}^{2}})={{c}^{2}}\].