Question

The pedal points of a perpendicular drawn from origin on the line $3 x+4 y-5=0$, is
A. $\left(\dfrac{3}{5}, 2\right)$
B. $\left(\dfrac{3}{5}, \dfrac{4}{5}\right)$
C. $\left(-\dfrac{3}{5},-\dfrac{4}{5}\right)$
D. $\left(\dfrac{30}{17}, \dfrac{19}{17}\right)$

Answer 1

Hint: Take the pedal point as $(h,k)$. We are given that the point is drawn from the origin. Therefore $({x_1},{y_1}) = (0,0)$. We can obtain the value of $a,b,c$ from the equation of the line given. If $(h,k)$ is the pedal point of the perpendicular from $({x_1},{y_1})$ to the line $ax + by + c = 0$ , then
$\dfrac{{\left( {h{\text{ }} - {\text{ }}{{\text{x}}_1}} \right)}}{a} = \dfrac{{\left( {k{\text{ }} - {\text{ }}{{\text{y}}_1}} \right)}}{b}{\text{ }} = {\text{ }}\dfrac{{ - \left( {a{x_1}\; + {\text{ b}}{{\text{y}}_1} + c} \right)}}{{\left( {a\; + {\text{ b}}} \right)}}$. By substituting for ${x_1},{y_1},a,b$ and $c$ we can obtain the values of $(h,k)$.

Complete step by step solution:
We are given that the equation of the perpendicular drawn from the origin is $3x + 4y - 5 = 0$.
Therefore, the pedal point lies on the line $3x + 4y - 5 = 0$.
Let the coordinate of the pedal point be $(h,k)$
If $(h,k)$ is the pedal point of the perpendicular from $({x_1},{y_1})$to the line $ax + by + c = 0$ , then
$\dfrac{{\left( {h{\text{ }} - {\text{ }}{{\text{x}}_1}} \right)}}{a} = \dfrac{{\left( {k{\text{ }} - {\text{ }}{{\text{y}}_1}} \right)}}{b}{\text{ }} = {\text{ }}\dfrac{{ - \left( {a{x_1}\; + {\text{ b}}{{\text{y}}_1} + c} \right)}}{{\left( {a\; + {\text{ b}}} \right)}}$
Here $({x_1},{y_1}) = (0,0)$ and$a = 3,b = 4,c = - 5$. Therefore, we have,
$\dfrac{{\left( {h{\text{ }} - {\text{ 0}}} \right)}}{3} = \dfrac{{\left( {k{\text{ }} - {\text{ 0}}} \right)}}{4}{\text{ }} = {\text{ }}\dfrac{{ - \left( {3(0)\; + {\text{ 4(0)}}\; - 5} \right)}}{{\left( {9\; + {\text{ 16}}} \right)}}$
$ \Rightarrow \dfrac{h}{3} = \dfrac{k}{4} = \dfrac{5}{{25}}$
$ \Rightarrow \dfrac{h}{3} = \dfrac{k}{4} = \dfrac{1}{5}$
$ \Rightarrow h = \dfrac{3}{5},k = \dfrac{4}{5}$
Therefore $(h,k) = \left( {\dfrac{3}{5},\dfrac{4}{5}} \right)$

Option ‘B’ is correct

Note: We can also find the value of $\left( {h,k} \right)$ by finding the slope of the line perpendicular to the given line and then applying the condition of perpendicular lines for slopes that is
${m_1}{m_2} = - 1$.