Answer: The correct option is (D) (4, -6).
Step-by-step explanation: Given that the areas of the triangles ADC and DCB are in the ratio 3 : 4.
We are to find the co-ordinates of point C.
From the diagram, we note that
the co-ordinates of point A and B are A(1, -9) and B(8,-2).
So, the length of the line segment AB, calculated by distance formula, is
![AB=\sqrt{(1-8)^2+(-9+2)^2}=\sqrt{49+49}=7\sqrt2.](https://tex.z-dn.net/?f=AB%3D%5Csqrt%7B%281-8%29%5E2%2B%28-9%2B2%29%5E2%7D%3D%5Csqrt%7B49%2B49%7D%3D7%5Csqrt2.)
Now, area of ΔADC is
![A_{ABC}=\dfrac{1}{2}\times AC\times CD,](https://tex.z-dn.net/?f=A_%7BABC%7D%3D%5Cdfrac%7B1%7D%7B2%7D%5Ctimes%20AC%5Ctimes%20CD%2C)
and area of ΔDCB is
![A_{DCE}=\dfrac{1}{2}\times BC\times CD.](https://tex.z-dn.net/?f=A_%7BDCE%7D%3D%5Cdfrac%7B1%7D%7B2%7D%5Ctimes%20BC%5Ctimes%20CD.)
According to the given information, we have
![A_{ADC}:A_{DCB}=3:4\\\\\\\Rightarrow \dfrac{A_{ADC}}{A_{DCE}}=\dfrac{3}{4}\\\\\\\Rightarrow \dfrac{\frac{1}{2}\times AC\times CD}{\frac{1}{2}\times BC\times CD}=\dfrac{3}{4}\\\\\\\Rightarrow \dfrac{AC}{BC}=3:4.](https://tex.z-dn.net/?f=A_%7BADC%7D%3AA_%7BDCB%7D%3D3%3A4%5C%5C%5C%5C%5C%5C%5CRightarrow%20%5Cdfrac%7BA_%7BADC%7D%7D%7BA_%7BDCE%7D%7D%3D%5Cdfrac%7B3%7D%7B4%7D%5C%5C%5C%5C%5C%5C%5CRightarrow%20%5Cdfrac%7B%5Cfrac%7B1%7D%7B2%7D%5Ctimes%20AC%5Ctimes%20CD%7D%7B%5Cfrac%7B1%7D%7B2%7D%5Ctimes%20BC%5Ctimes%20CD%7D%3D%5Cdfrac%7B3%7D%7B4%7D%5C%5C%5C%5C%5C%5C%5CRightarrow%20%5Cdfrac%7BAC%7D%7BBC%7D%3D3%3A4.)
So, the point C divides the line segment AB internally in the ratio 3 : 4.
We know that
if a point divides a line segment with end-points (a, b) and (c, d) internally in the ration m : n, then its co-ordinates are
![\left(\dfrac{mc+na}{m+n},\dfrac{md+nb}{m+n}\right).](https://tex.z-dn.net/?f=%5Cleft%28%5Cdfrac%7Bmc%2Bna%7D%7Bm%2Bn%7D%2C%5Cdfrac%7Bmd%2Bnb%7D%7Bm%2Bn%7D%5Cright%29.)
Since point C divides the line segment AB with end-points A(1, -9) and B(8, -2) internally, so the co-ordinates of point C will be
![\left(\dfrac{3\times 8+4\times 1}{3+4},\dfrac{3\times (-2)+4\times (-9)}{3+4}\right)\\\\\\=\left(\dfrac{28}{7},\dfrac{-42}{7}\right)\\\\\\=(4, -6).](https://tex.z-dn.net/?f=%5Cleft%28%5Cdfrac%7B3%5Ctimes%208%2B4%5Ctimes%201%7D%7B3%2B4%7D%2C%5Cdfrac%7B3%5Ctimes%20%28-2%29%2B4%5Ctimes%20%28-9%29%7D%7B3%2B4%7D%5Cright%29%5C%5C%5C%5C%5C%5C%3D%5Cleft%28%5Cdfrac%7B28%7D%7B7%7D%2C%5Cdfrac%7B-42%7D%7B7%7D%5Cright%29%5C%5C%5C%5C%5C%5C%3D%284%2C%20-6%29.)
Thus, the co-ordinates of point C are (4, -6).
Option (D) is correct.