Question

ANYONE SOLVE THIS QUESTION ...EXPLAIN ALL QUESTION...ND GIVE CORRECT ANSWER......
#AADI❤​

Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

The equation of any line in Slope - Intercept form is

$y = mx + c \: \: \: .....(1)$

Where m is the slope of the line (1)

Now the line (1) passes through ( 0, a)

So

$a = 0 + c$

$\therefore \: c \: = a$

So the equation (1) reduced to

$y = mx + a \: \: \: \:$

$\implies \: mx - y + a = 0 \: \: .....(2)$

Now the Perpendicular Distance from (2a 2a) is

$\displaystyle \: | \frac{2am - 2a + a}{ \sqrt{1 + {m}^{2} } } | = a$

$\implies\displaystyle \: | {2am - a | = a \sqrt{1 + {m}^{2} }}$

$\implies\displaystyle \: | {2m - 1 | = \sqrt{1 + {m}^{2} }}$

Squaring both sides

$4 {m}^{2} - 4m + 1 = 1 + {m}^{2}$

$\implies \: 3 {m}^{2} - 4m = 0$

$\implies \: m(3 {m} - 4) = 0$

So

$\displaystyle \: m = 0 \: , \: \: \frac{4}{3}$

Putting m = 0 in Equation (2)

$y \: = a \: \: \: .....(3)$

Putting

$\displaystyle \: m = \frac{4}{3}$

in Equation (2) we get

$\displaystyle \:y = \frac{4}{3} x + a$

$\implies \: 3y - 4x = 3a \: \: \: .....(4)$

Now the equation of a line perpendicular to Equation (4) is

$3x + 4y = k \: \:$

It passes through ( 2a, 2a)

So

$14a = k$

So the line is

$3x + 4y = 14a \: \: \: \: ......(5)$

Now the point of intersection of Equation (4) & (5) is obtained by solving them

So the point of intersection is

$\displaystyle \: ( \: \frac{6a}{5} , \frac{13a}{5} \: )$

This is one of the feet of the perpendicular

Again the another feet of the perpendicular on the line y = a from the point ( 2a, 2a ) is ( 2a, a )

Now the required line is the line joining the points

$\displaystyle \: ( \: \frac{6a}{5} , \frac{13a}{5} \: ) \: \: and \: \: ( \: 2a , a\: )$

So the required line is

$\displaystyle \: \frac{y - a}{x - 2a} = \frac{ \frac{13a}{5} - a}{ \frac{6a}{5} - 2a}$

$\implies \: \displaystyle \: \frac{y - a}{x - 2a} = \frac{ 8 a}{ - 4a}$

$\implies \: \displaystyle \: \frac{y - a}{x - 2a} = - 2$

$\implies \: y - a = - 2x + 4a$

$\implies \: y + 2x = 5a$

Hence the proof follows