Question

Three girls Rani, Mansi and Sneha are talking to each other and maintaining a social distance due to COVID-19. They are on the vertices of a triangle whose coordinates are Rani (2, -2), Mansi (1,1) and Sneha (-1,0).
The equation of line formed by Rani and Mansi is
3x-y = 4 b) 3x+y = 4 c) x-3y = 4 d) x+3y = 4
Slope of equation of line formed by Rani and Sneha is
b) c) d)
The equation of median through Rani is
a) 5x+4y = 2 b) 5x-4y = 2 c) 4x-5y = 1 d) None of these
The equation of altitude through Mansi is
3x-2y =1 b) 2x+3y = 5 c) x+2y = 3 d) None of these
The equation of line passing through Rani and parallel to line formed by Mansi and Sneha is
x-2y = 4 b) x+2y = 6 c) x-2y = 6 d) 2x+y = 4
​

Answer 1

Given:

Rani (2,-2)   Mansi (1,1)  Sneha (-1,0)

(i) The equation of the line formed by Rani and Mansi is

          (b) 3x+y = 4

We have Rani (2,-2)=(x₁y₁,)  and Mansi (1,1)=(x₂,y₂)

Slope of the line through Rani and Mansi is m

where  m = $\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

            m = $\frac{1-(-2)}{1-2}$

            m = -3

General equation of a line is y=mx+c

y= -3x+c

let us put x=2 and y= -2

-2= -3(2)+c

c=4

Now line equation is 3x+y =4

(ii)Slope of line through Rani and Sneha  is   -2

Rani (2,-2)  and Sneha(-1,0)

By using the slope formula

m= $\frac{0-2}{-1-(-2)}$

m = -2

so slope of line through Rani and Sneha  is -2

(iii)Median through Rani

      (a) 5x+4y = 2

We know that median through Rani passes through midpoint of line passing through Mansi and Sneha

Midpoint of Mansi and Sneha is $(\frac{1-1}{2} ,\frac{1+0}{2} )$ = $(0,\frac{1}{2})$

So now Median passes through points (2,-2) and (0, 1/2)

Slope of median is m= $\frac{\frac{1}{2}-(-2) }{0-2}$

                                m= $-\frac{5}{4}$

y = mx+c

y = (-5/4)x+c

put x=2 , y= -2

-2 = (-5/4)2 + c

c = 10/4 -2 = 10/4 - 8/4 = 2/4 = 1/2

c=1/2

So equation is y = (-5/4)x+1/2

4y = -5x+2

5x+4y =2

Answer 2

Question (i) :- The equation of the line formed by Rani and Mansi is

a) 3x-y =4 b) 3x +y =4 c)x-3y = 4 d)x +3y =4

Solution :-

given that,

• Rani (2,-2)
• Mansi (1,1)

so,

→ slope of line = (y2 - y1)/(x2 - x1) = 1 -(-2) / (1 - 2) = (1 + 2)/(-1) = (-3) .

then,

→ Equation of line = (y - y1) = m(x - x1) => y - (-2) = (-3)(x - 2) => y + 2 = -3x + 6 => y + 3x = 6 - 2 => 3x + y = 4 (b) (Ans.)

Question (i) :- Slope of the line formed by Rani and Sneha is :- a)-3/2 b) 2/3 c)-2/3 d) 1/3

Solution :-

given that,

• Rani (2,-2)
• Sneha (-1,0)

so,

→ slope of line = (y2 - y1)/(x2 - x1) => 0 -(-2) / (-1 - 2) => 2/(-3) = (-2/3) (c) (Ans.)

Question (iii) :- The equation of altitude through Mansi is :-

a) 2x+3y =5 b) 3x-2y = 1 c) None of these d)x+ 2y =3

Solution :-

given that,

• Mansi (1,1)
• Rani (2,-2)
• Sneha (-1,0)

so,

→ Slope of Rani and sneha = (-2/3) (solved in Question 2).

then,

→ Equation of altitude passing through mansi(1,1) vertex = y - y1 = m(x - x1) => y - 1 = (-2/3)(x - 1) => 3y - 3 = -2x + 2 => 3y + 2x = 2 + 3 => 2x + 3y = 5 (a) (Ans.)

Question (iv) :- The equation of median from Rani is :-

a) 5x+ 4y = 2 b) 5x 4y = 2 c) 4x- 5y = 1 d) None of these

Solution :-

given that,

• Mansi (1,1)
• Rani (2,-2)
• Sneha (-1,0)

since the median from Rani passes through midpoint of line passing through Mansi and Sneha .

→ Midpoint of Mansi and Sneha = (x1 + x2)/2 , (y1 + y2)/2 = (1 - 1)/2 = 0 and (0 + 1)/2 = (1/2) .

then,

→ Slope of midpoint(0,1/2) and Rani coordinates(2,-2) = (y2 - y1)/(x2 - x1) => (-2 - 1/2) / (2 - 0) => (-5/2) / 2 = (-5/4) .

therefore,

→ Equation of line = y - y1 = m(x - x1) => y - (-2) = (-5/4)(x - 2) => 4y + 8 = -5x + 10 => 5x + 4y = 2 (a) (Ans.)

Question (v) :- The equation of line passing through Rani and parallel to the line formed by Mansi and Sneha is :-

a) x-2y =4 b) x+2y = 6 c) x-2y = 6 d) 2x+y = 4

Solution :-

given that,

• Mansi (1,1)
• Rani (2,-2)
• Sneha (-1,0)

→ Slope of line formed by Mansi and Sneha = (y2 - y1)/(x2 - x1) => (0 - 1) / (-1 - 1) => (-1)/(-2) = (1/2) .

now, we know that, when lines are parallel their slope is equal .

then,

→ Slope of parallel line = (1/2) .

therefore,

→ Equation of line parallel and passing through Rani = y - y1 = m(x - x1) => y - (-2) = (1/2)(x - 2) => 2y + 4 = x - 2 => x - 2y = 6 (c) (Ans.)

Learn more :-

The orthocentre of the triangle with vertices A (1 , 3), B(2 , -1) and C ( 0 , -3)

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