In what ratio does the x …

We have to find the  ratio in which  the x-axis divides the line segment joining the points (- 4, - 6) and (- 1, 7).We will also  find the coordinates of the point of division.

Let x-axis  divides the line-segment joining (- 4, - 6) and (-1, 7) at the point P in the ratio 1: k.
Now, the coordinates of point of division P,
{tex}= \frac { 1 × ( - 1 ) + k× ( - 4 ) } { k + 1 } , \frac { 1 × 7 + k × ( - 6 ) } { k + 1 }{/tex}
{tex}= \frac { - 1 - 4 k } { k + 1 } , \frac { 7 - 6 k } { k + 1 }{/tex}
Since P lies on x-axis, therefore
{tex}\frac { 7 - 6 k } { k + 1 } = 0{/tex}
or, 7 - 6k = 0 {tex}\Rightarrow k = \frac { 7 } { 6 }{/tex}
Hence, the ratio is 1:{tex}\frac 76{/tex} or 6 : 7
And the coordinates of P are {tex}\frac{-1-\frac{28}{6}}{\frac{7+6}{6}},0=\frac{-6-28}{7+6},0{/tex}

i.e {tex}\left( - \frac { 34 } { 13 } , 0 \right){/tex}.