Question

A line AB meets x-axis at A and y-axis at B point P(4 - 1) divides AB in the ratio 1 : 2 .
i) Find the coordinates of A and B
ii) Find the equation of the line through P and perpendicular to AB ​

Answer 1

$\huge\mathrm{Answer ⭐}$

Given

✍️ Ratio of AB is 1 : 2

✍️ P coordinate is (4 , -1 )

To find

✍️ Coordinates of A and B

✍️ Equation of line through P and perpendicular to AB

Solution

★

Let the coordinates of point A be - ( a , 0)

Let the coordinates of point B be - ( 0 , b )

According to the question,

Point P(4,-1) divides the line segment AB in the ratio 1 : 2 , now the coordinates of P are -

( (2 × a + 1 × 0 )/ (2+1 ) ) ,( ( 2 × 0 + 1 × b ) / (2+1) ) = ( 4 , -1 )

➡ ( 2a/3 , b/3 ) = ( 4 , -1 )

2a/3 = 4

➡ a = 6

b/3 = -1

➡ b = -3

Thus, A's coordinate is ( a , 0 ) = ( 6 , 0 )

B's coordinate is ( b , 0 ) = ( 0 , -3 )

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Slope of AB = ( -3-0 ) / ( 0-6 )

➡ Slope of AB = -3 / -6

➡ Slope of AB = 1 / 2 = m

m × m1 = - 1 [ rule of perpendicularity of line segment ]

➡ 1/2 × m1 = -1

➡ m1 = -2

Therefore, The equation of line through P(4, -1) is -

y - ( -1 ) = - 2 ( x - 4 )

➡ y + 1 = -2x + 8

➡ 2x + y - 7 = 0

The line of equation of AB is 2x + y = 7

Answer 2

Step-by-step explanation:

Let the coordinates of the points A and B be (a,0) and (0,b) respectively.

(i)

As the point P(4,-1) divides the line segment AB in the ratio 1:2 i.e. $\sf{\dfrac{AP}{PB}=\dfrac{1}{2}}$, the coordinates of the point P are,

$\implies\sf{\dfrac{(2a+1\times\:0)}{(2+1)},\dfrac{(2\times\:0+1\times\:b)}{(2+1)}}$

$\implies\sf{(\dfrac{2a}{3},\dfrac{b}{3})}$

But P is (4,-1)

2a/3=4

$\implies$ 2a = 12

$\implies$ a = 6

&

b/3=-1

$\implies$b = -3

Hence, the coordinates of the points A and B are (6,0) and (0,-3) respectively.

_______________________

(ii)

Slope of the line AB =$\sf\dfrac{ (-3-0)}{(0-6)}$

→ Slope of the line AB = ½

$\therefore$ The slope of a line perpendicular to AB = -2

$\therefore$ The equation of the line through P (4,-1) and perpendicular to the line AB is ,

y -(-1) = -2(x-4)

→ y+1 = -2x +8

→ 2x+y-7=0