SATHEE: Straight Line and Pair of Straight Lines 1 Question 14

Straight Line and Pair of Straight Lines 1 Question 14

14. Two sides of a parallelogram are along the lines, $x + y = 3$ and $x - y + 3 = 0$ . If its diagonals intersect at $(2$ , $4)$ , then one of its vertex is

(2019 Main, 10 Jan II)

(a) $(3, 6)$

(b) $(2, 6)$

(d) $(3, 5)$

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Solution:

According to given information, we have the following figure

[Note that given lines are perpendicular to each other as $m_{1} \times m_{2} = - 1]$

Clearly, point $A$ is point of intersection of lines

and $\begin{aligned} x + y & = 3 x - y & = - 3 \end{aligned}$

So, $A = (0, 3)$ [solving Eqs. (i) and (ii)]

Now, as point $M (2, 4)$ is mid-point of line joining the points $A$ and $C$ , so

$\begin{aligned} (2, 4) = \frac{0 + x_{2}}{2}, & \frac{3 + y_{2}}{2} \\ ∵ mid- point = \frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2} \end{aligned}$

$\begin{array}{lll} \Rightarrow & 2 & = \frac{0 + x_{2}}{2}; 4 = \frac{3 + y_{2}}{2} \\ \Rightarrow & x_{2} = 4 and y_{2} = 5 \\ ∴ Thus, & C & \equiv (4, 5) \end{array}$

Now, equation of line $B C$ is given by

$\begin{aligned} (y - y_{1}) & = m (x - x_{1}) \\ y - 5 & = 1 (x - 4) \end{aligned}$

[line $B C$ is parallel to $x - y + 3 = 0$ and slope

$\begin{aligned} of x - y + 3 = 0 is \frac{(- 1)}{(- 1)} = 1] \\ \Rightarrow y = x + 1 \end{aligned}$

and equation of line $D C$ is

$\begin{aligned} y - 5 = - 1 (x - 4) \\ [line D C is parallel to x + y = 3 and \\ slope of x + y = 3 is \frac{- 1}{1} = - 1] \end{aligned}$

$\Rightarrow x + y = 9$ …(iv)

On solving Eqs. (i) and (iii), we get $B (1, 2)$ and on solving Eqs. (ii) and (iv), we get $D (3, 6)$

Straight Line and Pair of Straight Lines 1 Question 14

14. Two sides of a parallelogram are along the lines, $x + y = 3$ and $x - y + 3 = 0$ . If its diagonals intersect at $(2$ , $4)$ , then one of its vertex is

Solution:

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Straight Line and Pair of Straight Lines 1 Question 14

14. Two sides of a parallelogram are along the lines, x+y=3 and x−y+3=0. If its diagonals intersect at (2, 4), then one of its vertex is

Solution:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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14. Two sides of a parallelogram are along the lines, $x + y = 3$ and $x - y + 3 = 0$ . If its diagonals intersect at $(2$ , $4)$ , then one of its vertex is