Consider the following statements:

1) The length p of the perpendicular form the origin to the line ax + by = c satisfies the origin to the relation \({{\rm{p}}^2} = \frac{{{{\rm{c}}^2}}}{{{{\rm{a}}^2} + {{\rm{b}}^2}}}\)

2) The length p the perpendicular from the origin to the line \(\frac{{\rm{x}}}{{\rm{a}}} + \frac{{\rm{y}}}{{\rm{b}}} = 1\) satisfies the relation \(\frac{1}{{{{\rm{p}}^2}}} = \frac{1}{{{{\rm{a}}^2}}} + \frac{1}{{{{\rm{b}}^2}}}\)

3) The length p of the perpendicular from the origin to the line y = mx + c satisfies the relation \(\frac{1}{{{{\rm{p}}^2}}} = \frac{{1 + {{\rm{m}}^2} + {{\rm{c}}^2}}}{{{{\rm{c}}^2}}}\)

Which of the above is/are correct?This question was previously asked in

NDA (Held On: 22 April 2018) Maths Previous Year paper

Option 3 : 1 and 2 only

Electric charges and coulomb's law (Basic)

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10 Questions
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__Concept:__

**Perpendicular distance **from a point on a line**:**

Let a point be (x_{1}, y_{1}) and equation of line be ax + by + c = 0 then:

Perpendicular distance \(= \frac{{\left| {{\rm{a}}{{\rm{x}}_1}{\rm{\;}} + {\rm{\;b}}{{\rm{y}}_1}{\rm{\;}} + {\rm{\;c}}} \right|}}{{\sqrt {{{\rm{a}}^2}{\rm{\;}} + {\rm{\;}}{{\rm{b}}^2}} }}\)

__Calculation:__

For **statement (1),**

Point is (0,0) and equation of line is ax + by - c = 0.

So, \({\rm{p}} = \frac{{\left| { - {\rm{c}}} \right|}}{{\sqrt {{{\rm{a}}^{2{\rm{\;}}}} + {\rm{\;}}{{\rm{b}}^2}} }}\)

\(\Rightarrow {\rm{\;p}} = \frac{{\rm{c}}}{{\sqrt {{{\rm{a}}^{2{\rm{\;}}}} + {\rm{\;}}{{\rm{b}}^2}} }}\)

Squaring both sides

\(\Rightarrow {{\rm{p}}^2} = \frac{{{{\rm{c}}^2}}}{{{{\rm{a}}^2} + {{\rm{b}}^2}}}\)

So, **first statement **is correct.

For **statement (2),**

Point is (0,0) and equation of line is \(\frac{{\rm{x}}}{{\rm{a}}} + \frac{{\rm{y}}}{{\rm{b}}} = 1\)

Equation of line will be bx + ay = ab

⇒ bx + ay - ab = 0

So, \({\rm{p}} = \frac{{\left| { - {\rm{ab}}} \right|}}{{\sqrt {{{\rm{b}}^2}{\rm{\;}} + {\rm{\;}}{{\rm{a}}^2}} }}\)

\(\Rightarrow {\rm{p}} = \frac{{{\rm{ab}}}}{{\sqrt {{{\rm{b}}^2}{\rm{\;}} + {\rm{\;}}{{\rm{a}}^2}} }}\)

Squaring both sides

\(\Rightarrow {{\rm{p}}^2} = \frac{{{{\rm{a}}^2}{{\rm{b}}^2}}}{{{{\rm{a}}^2} + {{\rm{b}}^2}}}\)

Taking reciprocal of both sides,

\(\Rightarrow \frac{1}{{{{\rm{p}}^2}}} = \frac{{{{\rm{a}}^2} + {{\rm{b}}^2}}}{{{{\rm{a}}^2}{{\rm{b}}^2}}}\)

\(\Rightarrow \frac{1}{{{{\rm{p}}^2}}} = \frac{1}{{{{\rm{b}}^2}}} + \frac{1}{{{{\rm{a}}^2}}}\)

So, **second statement **is correct.

For **statement (3),**

Point is (0,0) and equation of line is y - mx - c = 0

So, \({\rm{p}} = \frac{{\left| { - {\rm{c}}} \right|}}{{\sqrt {{{\rm{m}}^2} + {1^2}} }}\)

Squaring both sides,

\(\Rightarrow {{\rm{p}}^2} = \frac{{{{\rm{c}}^2}}}{{{{\rm{m}}^2} + 1}}\)

Taking reciprocal of both sides,

\(\Rightarrow \frac{1}{{{{\rm{p}}^2}}} = \frac{{{{\rm{m}}^2} + 1}}{{{{\rm{c}}^2}}}\)

So,