Question

The coordinate of the foot of the perpendicular from $({x_1},{y_1})$ to the line $ax + by + c = 0$ are.
A) $\left( {\dfrac{{{b^2}{x_1} - ab{y_1} - ac}}{{{a^2} + {b^2}}},\dfrac{{{b^2}{y_1} - ab{x_1} - bc}}{{{a^2} + {b^2}}}} \right)$
B) $\left( {\dfrac{{{b^2}{x_1} + ab{y_1} + ac}}{{{a^2} + {b^2}}},\dfrac{{{a^2}{y_1} + ab{x_1} + bc}}{{{a^2} + {b^2}}}} \right)$
C) $\left( {\dfrac{{a{x_1} + b{y_1} + ab}}{{a + b}},\dfrac{{a{x_1} - b{y_1} - ab}}{{a + b}}} \right)$
D) None

Answer 1

Hint:The point on a triangle’s leg, opposite a certain vertex where the perpendicular that passes through that vertex crosses the side is known as the perpendicular foot or foot of an altitude. For the given, firstly find the slope of that particular line and then use the condition for perpendicular lines’ slopes i.e., ${m_1}{m_2} = - 1$.

Formula Used: We have the equation of line $ax + by + c = 0$ . To find the perpendicular distance from the given point $({x_1},{y_1})$ we have,
$d = \left| {\dfrac{{a{x_1} + b{y_1} + c}}{{\sqrt {{a^2} + {b^2}} }}} \right|$

Complete step by step Solution:
We have the equation of line $ax + by + c = 0$ . To find the perpendicular distance from the given point $({x_1},{y_1})$ we have,
$d = \left| {\dfrac{{a{x_1} + b{y_1} + c}}{{\sqrt {{a^2} + {b^2}} }}} \right|$
To find the coordinates of the foot of the perpendicular from the point $({x_1},{y_1})$ to the line $ax + by + c = 0$ , we have
$\dfrac{{h - {x_1}}}{a} = \dfrac{{k - {y_1}}}{b} = \dfrac{{ - a{x_1} + b{y_1} + c}}{{{a^2} + {b^2}}}$
Solving this equation to find the value of $h$, we get
$\dfrac{{h - {x_1}}}{a} = \dfrac{{ - a{x_1} + b{y_1} + c}}{{{a^2} + {b^2}}}$
$ \Rightarrow h = \left( {\dfrac{{{b^2}{x_1} - ab{y_1} - ac}}{{{a^2} + {b^2}}}} \right)$
Similarly, to find $k$, we take
$\dfrac{{k - {y_1}}}{b} = \dfrac{{ - a{x_1} + b{y_1} + c}}{{{a^2} + {b^2}}}$
$ \Rightarrow k = \dfrac{{{b^2}{y_1} - ab{x_1} - bc}}{{{a^2} + {b^2}}}$
Therefore, we have the coordinates of the foot of the perpendicular
$\left( {\dfrac{{{b^2}{x_1} - ab{y_1} - ac}}{{{a^2} + {b^2}}},\dfrac{{{b^2}{y_1} - ab{x_1} - bc}}{{{a^2} + {b^2}}}} \right)$

Therefore, the correct option is (A).

Note: If we have two equation of lines $y = {m_1}x + {c_1}$ and ${y_2} = {m_2}x + {c_2}$ which are at an angle ${90^\circ }$ i.e., perpendicular to one another, then the relation between the slopes of these two lines is given as ${m_1}{m_2} = - 1$ . This relation is also used to find whether the given lines are perpendicular or not, hence this relation is called the condition for perpendicularity.