Question

Show that the perpendicular drawn from the point (4,1) on the line joining (6,5) and (2,-1) divides it in the ratio 8:5

Answer 1

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Answer 2

Answer:

Slope of line  line joining (6,5) and (2,-1)

$m = \frac{-1-5}{2-6}=\frac{3}{2}$

Since we know that the product of slopes of perpendicular lines is -1

So, $\frac{3}{2}m = - 1$

$m = -\frac{2}{3}$

So. the slope of perpendicular line is -2/3

Equation of perpendicular line =$y-y_1=m(x-x_1) = y-1=\frac{-2}{3}(x-4)$

Let  the perpendicular drawn from the point (4,1) on the line joining (6,5) and (2,-1) divides it in the ratio 8:5

So, to find point of division

$x=\frac{mx_2+nx_1}{m+n} , y =\frac{my_2+ny_1}{m+n}$

$x=\frac{8(2)+5(6)}{8+5} , y =\frac{8(-1)+5(5)}{8+5}$

$x=\frac{46}{13}, y =\frac{17}{13}$

Now this point must satisfy the equation of perpendicular

$\frac{4}{13}=\frac{4}{13}$

Hence proved that  the perpendicular drawn from the point (4,1) on the line joining (6,5) and (2,-1) divides it in the ratio 8:5