Question

The number of straight lines which can be drawn through the point $ (-2,\,\,\,2) $ so that its distance from $ (-3,\,\,1) $ will be equal $6$ units is

• A. one
• B. two
• C. infinite
• D. zero

Answer 1

The Correct Option is D

Let the slope be m of the straight line which passes through
$ (-2,2), $ then equation $ (y-2)=m(x+2) $
$ mx-y+(2m+2)=0 $ ..(i)
$ \because $ Perpendicular distance of line (i) from point (given) $ (3,-1)=6 $
$ \Rightarrow $ $ \frac{|m(3)-(-1)+(12m+2)|}{\sqrt{{{m}^{2}}+{{(-1)}^{2}}}}=6 $
$ |3m+1+2m+2|=6\sqrt{{{m}^{2}}+1} $
Squaring on both sides,
$ \Rightarrow $ $ {{(5m+3)}^{2}}=36({{m}^{2}}+1) $
$ \Rightarrow $ $ 25{{m}^{2}}+9+30m=36{{m}^{2}}+36 $
$ \Rightarrow $ $ 11{{m}^{2}}-30m+27=0 $ ..(ii)
Now, $ \Delta ={{B}^{2}}-4AC=900-4(11)\,(27) < 0 $
$ \because $ Discriminate of E (ii) is negative. i.e., slope of the given line is imaginary. So, no line drawn through the point $ (2,2) $ .