A) \[AP=\frac{1}{3}AB\]
B) \[AP=PB\]
C) \[PB=\frac{1}{3}AB\]
D) \[AP=\frac{1}{2}AB\]
Correct Answer: D
Solution :
| [d] Given that, the point \[P(2,1)\] lies on the line segment joining the points \[A(4,2)\] and \[B(8,4)\]. |
| Now, distance between \[A(4,2)\] and \[P(2,1)\]. |
| \[AP=\sqrt{{{(2-4)}^{2}}+{{(1-2)}^{2}}}=\sqrt{4+1}=\sqrt{5}\] |
| Distance between \[A(4,2)\]and \[B(8,4)\] |
| \[AB=\sqrt{{{(8-4)}^{2}}+{{(4-2)}^{2}}}\] |
| \[=\sqrt{{{(4)}^{2}}+{{(2)}^{2}}}=\sqrt{16+4}=\sqrt{20}=2\sqrt{5}\] |
| Distance between \[B(8,4)\] and \[P(2,1),\] |
| \[BP=\sqrt{{{(8-2)}^{2}}+{{(4-1)}^{2}}}=\sqrt{{{6}^{2}}+{{3}^{2}}}\] |
| \[=\sqrt{36+9}=\sqrt{45}=3\sqrt{5}\] |
| \[\therefore \,\,\,\,\,\,\,\,\,\,\,\,\,\,AB=2\sqrt{5}=2AP\,\,\Rightarrow \,\,AP=\frac{AB}{2}\] |
| Hence, required condition is \[AP=\frac{AB}{2}\] |
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