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  • The coordinates of the foot of perpendicular from the point P(2,3) to the line <div class='qna

The coordinates of the foot of perpendicular from the point P(2,3) to the line 3 x-4 y-16=0

Option: 1

(5,-1)


Option: 2

(3,4)


Option: 3

(1,2)


Option: 4

(2,3)


Answers (1)

best_answer

\mathbf{\frac{x_{2}-x_{1}}{a}=\frac{y_{2}-y_{1}}{b}=-\frac{\left(a x_{1}+by_{1}+c\right)}{\left(a^{2}+b^{2}\right)}}

Applying this formula

\\\frac{x_2-2}{3}=\frac{y_2-3}{-4}=-\frac{(3(2)-4(3)-19)}{3^2+4^2}\\\\ \frac{x_2-2}{3}=\frac{y_2-3}{-4}=1\\\\ \frac{x_2-2}{3}=1\,\,\,\,and\,\,\,\frac{y_2-3}{-4}=1 \\\\ x_2=5\,\,\,and\,\,\,y_2=-1

 

Method 2

\\\text{Let M(a,b) be the foot of perpendicular}\\\text{So, it lies on the line }3 x-4 y-19=0....(i)\\\Rightarrow 3a-4b-19=0\\ \\\text{Also, Point PM is perpendicular to this line }\\ \Rightarrow \frac{b-3}{a-2} \times \frac{3}{4}=-1\\ \Rightarrow 3b+4a-17=0....(ii)\\\\ 3\times \text{Equation (i) + }4\times equation(ii)\\ \Rightarrow 9a-12b-57+12b+16a-68=0\\ \Rightarrow 25a=125\\\Rightarrow a=5\\\Rightarrow 3b=17-4*5\\ \Rightarrow b=-1\\

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Pankaj

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