A) \[\frac{1}{{{p}^{2}}}={{a}^{2}}+{{b}^{2}}\]
B) \[{{p}^{2}}={{a}^{2}}+{{b}^{2}}\]
C) \[{{p}^{2}}=\frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}\]
D) \[\frac{1}{{{p}^{2}}}=\frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}\]
Correct Answer: D
Solution :
If intercepts on the axes are a and b, then equation of line is \[\frac{x}{a}+\frac{y}{b}=1\] \[\Rightarrow \] \[bx+ay=ab\] ...(i) Length of perpendicular drawn on line (i) from the point (0,0). \[p=\left| \frac{0+0-ab}{\sqrt{{{a}^{2}}+{{b}^{2}}}} \right|\] \[\Rightarrow \] \[p=\frac{ab}{\sqrt{{{a}^{2}}+{{b}^{2}}}}\] \[\Rightarrow \] \[p=\frac{{{a}^{2}}{{b}^{2}}}{{{a}^{2}}+{{b}^{2}}}\] \[\Rightarrow \] \[\frac{1}{{{p}^{2}}}=\frac{{{a}^{2}}+{{b}^{2}}}{{{a}^{2}}{{b}^{2}}}\] \[\Rightarrow \] \[\frac{1}{{{p}^{2}}}=\frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}\]
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