A) \[(-4,-3)\]
B) \[(3,-4)\]
C) \[(-3,4)\]
D) \[(-3,-4)\]
Correct Answer: C
Solution :
Let\[P({{x}_{1}},{{y}_{1}})\]is the image of the point\[Q(4,-3)\]. Midpoint of\[PQ\left( \frac{{{x}_{1}}+4}{2},\frac{{{y}_{1}}-3}{2} \right)\]. This pointlies on\[y=x\]. \[\therefore \] \[\frac{{{x}_{1}}+4}{2}=\frac{{{y}_{1}}-3}{2}\] \[\Rightarrow \] \[{{x}_{1}}-{{y}_{1}}=-7\] ?(i) Slope of \[PQ=\frac{-3-{{y}_{1}}}{4-{{x}_{1}}}\] and slope of\[y=x\]is 1 Since, PQ is perpendicular to\[y=x\] \[\therefore \] \[\left( \frac{-3-{{y}_{1}}}{4-{{x}_{1}}} \right).1=-1\] \[\Rightarrow \] \[-3-{{y}_{1}}=-4+{{x}_{1}}\] \[\Rightarrow \] \[{{x}_{1}}+{{y}_{1}}=1\] ...(ii) On solving Eqs. (i) and (ii), \[{{x}_{1}}=-3,{{y}_{1}}=4\] Hence, required point is\[(-3,4)\].
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