Question

find the coordinates of the foot of prependicular drawn from the point (3,4) on the line x+y-2=0​

Answer 1

GIVEN.

foot of perpendicular drawn from the point

=> ( 3,4) on the line x + y - 2 = 0

EXPLANATION.

equation of line = x + y - 2 = 0

equation of the curve = y = mx + c

=> y = 2 - x

=> y = -x + 2

slope of the line = -1

=> point are = ( 3,4)

Let we assume that ( a, b) are coordinate of

foot of perpendicular.

then equation will be written as = a + b - 2 = 0

as we know that,

=> M1 X M2 = -1

=> -1 X M2 = -1

=> M2 = 1

equation of perpendicular

=> ( y - Y1) = m ( x - X1)

=> ( b - 4 ) = 1 ( a - 3 )

=> b - 4 = a - 3

=> b - a = 1 .....(2)

From equation (1) and (2) we get,

=> a + b = 2 .....(1)

=> b - a = 1 .....(2)

we get,

=> 2b = 3

=> b = 3/2

put the value of b = 3/2 in equation (1)

we get,

=> a + 3/2 = 2

=> a = 2 - 3/2

=> a = 4 - 3 / 2

=> a = 1/2

Therefore,

The coordinate of foot of perpendicular

drawn from the point ( 3,4)

=> ( 1/2 , 3/2 )

Answer 2

$\huge\mathtt{Question}$

find the coordinates of the foot of prependicular drawn from the point (3,4) on the line x+y-2=0

$\huge\mathtt{Given}$

Tʜᴇ ғᴏᴏᴛ ᴏғ ᴘᴇʀᴍᴇɴᴅɪᴄᴜʟᴀʀ ᴅʀᴀᴡɴ ғʀᴏᴍ ᴛʜᴇ ᴘᴏɪɴᴛ

➪ (3,4) ɪɴ ᴛʜᴇ ʟɪɴᴇ x + ʏ - 2 = 0

$\huge\mathtt{Solution}$

ᴇǫᴜᴀᴛɪᴏɴ ᴏғ ʟɪɴᴇ = x + ʏ - 2 = 0

ᴇǫᴜᴀᴛɪᴏɴ ᴏғ ᴛʜᴇ ᴄᴜʀᴠᴇ = ʏ = ᴍx + ᴄ

→ ʏ = 2 - x

→ ʏ = -x + 2

Sʟᴏᴘᴇ ᴏғ ᴛʜᴇ ʟɪɴᴇ = -1

→ ᴘᴏɪɴᴛs ᴀʀᴇ = (3,4)

ɴᴏᴡ, ʟᴇᴛ ᴜs ᴀssᴜᴍᴇ ᴛʜᴀᴛ (ᴀ,ʙ) ᴀʀᴇ ᴄᴏᴏʀᴅɪɴᴀᴛᴇ ᴏғ ғᴏᴏᴛ ᴏғ ᴘᴇʀᴘᴇɴᴅɪᴄᴜʟᴀʀ.

ᴛʜᴇɴ, ᴇǫ ᴡɪʟʟ ʙᴇ ᴡʀɪᴛᴛᴇɴ ᴀs ᴀ + ʙ - 2 = 0

ᴀs ᴡᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

→ ᴍ1 × ᴍ2 = -1

→ ᴍ2 = 1

Eǫᴜᴀᴛɪᴏɴ ᴏғ ᴘᴇʀᴘᴇɴᴅɪᴄᴜʟᴀʀ,

→ (ʏ-ʏ1) = ᴍ(x-x1)

→ ʙ - 4 = 1 ( ᴀ-3)

→ ʙ - 4 = ( ᴀ-3)

→ ʙ - ᴀ = 1........................................................(2)

ғʀᴏᴍ ᴇǫ 1 ᴀɴᴅ 2, ᴡᴇ ɢᴇᴛ,

→ ᴀ + ʙ = 2....................................(1)

→ ʙ - ᴀ = 1......................................(2)

ғʀᴏᴍ ᴀʙᴏᴠᴇ ᴡᴇ ɢᴇᴛ,

→ 2ʙ = 3

→ ʙ = $\frac{3}{2}$

Pᴜᴛᴛɪɴɢ ᴛʜᴇ ᴠᴀʟᴜᴇ ᴏғ ʙ = $\frac{3}{2}$ ɪɴ ᴇǫ (1) ᴡᴇ ɢᴇᴛ,

→ᴀ + $\frac{3}{2}$ = 2

→ᴀ = 2 - $\frac{3}{2}$

→ᴀ = 4 - $\frac{3}{2}$

→ᴀ = $\frac{1}{2}$

ᴛʜᴇʀᴇғᴏʀᴇ, ᴛʜᴇ ᴄᴏᴏʀᴅɪɴᴀᴛᴇ ᴏғ ᴛʜᴇ ғᴏᴏᴛ ᴏғ ᴘᴇʀᴘᴇɴᴅɪᴄᴜʟᴀʀ ᴅʀᴀᴡɴ ғʀᴏɴ ᴛʜᴇ ᴘᴏɪɴᴛ (3,4)

$\huge\mathtt{Answer}$

ᴛʜᴇ ᴛᴡᴏ ᴘᴏɪɴᴛs ᴀʀᴇ $\frac{1}{2}$ ᴀɴᴅ $\frac{3}{2}$.