question_answer

A line \[y=mx+1\] intersects the circle \[{{(x-3)}^{2}}+\,{{(y+2)}^{2}}=25\] at the points P and Q. If the midpoint of the line segment PQ has x-coordinate \[\frac{-\,3}{5}\] then which one of the following options is correct?

A) \[-\,3\,\,\le \,\,m\,\,<-1\]              

B) \[6\,\,\le \,\,m\,\,<\,\,8\]

C) \[4\,\,\le \,\,m\,\,<\,\,6\]              

D) \[2\,\,\le \,\,m\,\,<\,\,4\]

Correct Answer: D

Solution :

\[{{m}_{\text{AB}}}.\,{{m}_{\text{cm}}}\,=\,-\,1\]

\[\Rightarrow \]   \[m.\left( \frac{1\,-\,\frac{3}{5}\,m\,+2}{-\,\frac{3}{5}\,-\,3} \right)\,=\,-1\]

\[\Rightarrow \]   \[m\left( \frac{15-3m}{-\,18}\, \right)\,=\,-1\]

\[\Rightarrow \]   \[15\,-\,3{{m}^{2}}-18\,=\,0\]

\[\Rightarrow \]   \[{{m}^{2}}\,-\,5m\,+\,6\,=\,0\]

\[M=2,m=3\]\[\Rightarrow \] \[2\le ~m<4.\]