First slide

Equation of line in Straight lines

Question

Let $A (7, 5)$ and $B (1, 2)$ are any two points. The ratio in which the foot of the perpendicular from $P (2, 3)$ to the line $A B$ divides it is

Moderate

A

$3 : 1$
B

$1 : 3$
C

$4 : 1$
D

$1 : 4$

Solution

$The slope of A B is m = \frac{2 - 5}{1 - 7} = \frac{1}{2}$
$The equation of the line perpendicular to A B and passing through the point P (2, 3)_{i}$
$\begin{matrix} y - 3 = - 2 (x - 2) \\ y - 3 = - 2 x + 4 \\ 2 x + y - 7 = 0 \end{matrix}$
$The ratio of the line segment joining two points A, B divided by the line L = 0 is - \frac{L_{A}}{L_{B}}$
$Hence, the ratio of line segment joining points A (7, 5) and B (1, 2) is divided by the line$
$2 x + y - 7 = 0 is - \frac{L_{A}}{L_{s}} = - \frac{2 (7) + 5 - 7}{2 (1) + 2 - 7} = \frac{12}{3} = 4$
$Therefore, the ratio is 4 : 1$

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