Question

A line L passes through P(1, 2) such that P bisects the line segment intercepted between the axes, then the perpendicular distance of L from the origin is

Answer 1

SOLUTION

TO DETERMINE

The perpendicular distance from origin to the line passing through P (1, 2) such that P bisects the part intercepted between the axes

EVALUATION

Let the equation of the line is

$\displaystyle \sf{ \frac{x}{a} + \frac{y}{b} = 1}$

Now the line intersects x axis at A(a, 0) and y axis at B(0,b)

So the midpoint of the line joining the points A and B is

$\displaystyle \sf{ = \bigg(\frac{a + 0}{2} \: \: , \: \frac{0 + b}{2} \bigg)}$

$\displaystyle \sf{ = \bigg(\frac{a }{2} \: \: , \: \frac{b}{2} \bigg)}$

Now by the given condition

$\displaystyle \sf{ \frac{a }{2} = 1 \: \: , \: \frac{b}{2} = 2}$

$\implies \: \displaystyle \sf{ a = 2 \: \: , \: b = 4}$

So the equation of the line is

$\displaystyle \sf{ \frac{x}{2} + \frac{y}{4} = 1}$

$\displaystyle \sf{ \implies \: 2x + y = 4}$

So the required perpendicular distance from origin

$\displaystyle \sf{ = \bigg | \frac{4}{ \sqrt{ {2}^{2} + {1}^{2} } } \bigg| \: \: \: unit }$

$\displaystyle \sf{ = \frac{4}{ \sqrt{ 5 } } \: \: unit }$

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