Question

The nearest point on the line $3x+4y=25$ from the origin is

• A. $(-5,5)$
• B. $(3,4)$
• C. $(3,5)$
• D. $(3,-4)$

Answer 1

Answer: (B)

$(3,4)$

The nearest point from the origin, on the line $3x+4y=25$ will be the point where the perpendicular from the origi to the line $3x+4y=2x,$ will intersect this line.

For example $OP$ is the perpendicular from $O$ to $OP$.

$\therefore P$ is the nearest point from the origin to the given line.

$\because$ slope of the line $3x+4y=25$ is $-\frac{3}{4}$

$\therefore$ slope of the line perpendicular to this line $=\frac{4}{3}$

and perpendicular is drawn from $(0,0)$

$\therefore$ equation of the perpendicular will be

$y-0=\frac{4}{3}(x-0)\Rightarrow 4x-3y=0$

Now solving $3x+4y=25$ and $4x-3y=0,$ we get

$x=3$ and $y=4$

$\therefore$ The required point is $(3,4)$