First slide

Equation of line in Straight lines

Question

The point on the line $3 x - 2 y = 1$ which is closest to the origin is

Easy

A

$(\frac{3}{13}, - \frac{2}{13})$
B

$(\frac{5}{11}, \frac{2}{11})$
C

$(\frac{3}{5}, \frac{2}{5})$
D

None of these

Solution

The point can be taken as $(x, \frac{3 x - 1}{2})$
Let $z = x^{2} + \frac{{(3 x - 1)}^{2}}{4}$
[where $z =$ Square of the distance of the point from origin]
$z$ is min. for $x = \frac{3}{13}$
$∴$ The point is $(\frac{3}{13}, - \frac{2}{13})$
$s e c o n d m e t h o d : N e a r e s t p o i n t f r o m (0, 0) t o 3 x - 2 y - 1 = 0 i s e q u a l t o f o o t o f t h e p e r p e n d i c u l a r$
$d r a w n f r o m (0, 0) t o t h e l i n e . L e t (h, k) b e t h e f o o t o f o r i g i n o n t h e g i v e n l i n e . t h e r e f o r e \frac{h - 0}{3} = \frac{k - 0}{- 2} = - \frac{- 1}{3^{2} + 2^{2}} \Rightarrow h = \frac{3}{13} a n d k = \frac{- 2}{13}$

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