SATHEE: Straight Line and Pair of Straight Lines 1 Question 63

Straight Line and Pair of Straight Lines 1 Question 63

63. Two sides of a rhombus $A B C D$ are parallel to the lines $y = x + 2$ and $y = 7 x + 3$ . If the diagonals of the rhombus intersect at the point $(1, 2)$ and the vertex $A$ is on the $Y$ -axis, find possible coordinates of $A$ .

$(1985, 5 M)$

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Solution:

Let the coordinates of $A$ be $(0, α)$ . Since, the sides $A B$ and $A D$ are parallel to the lines $y = x + 2$ and $y = 7 x + 3$ , respectively.

$∴$ The diagonal $A C$ is parallel to the bisector of the angle between these two lines. The equation of the bisectors are given by

$\begin{aligned} \frac{x - y + 2}{\sqrt{2}} & = \pm \frac{7 x - y + 3}{\sqrt{50}} \\ \Rightarrow & 5 (x - y + 2) & = \pm (7 x - y + 3) \\ \Rightarrow & 2 x + 4 y - 7 = 0 & and 12 x - 6 y + 13 = 0 . \end{aligned}$

Thus, the diagonals of the rhombus are parallel to the lines $2 x + 4 y - 7 = 0$ and $12 x - 6 y + 13 = 0$ .

$\begin{aligned} Slope of A E & = - \frac{2}{4} or & \frac{12}{6} \\ \Rightarrow & \frac{2 - α}{1 - 0} & = - \frac{1}{2} or & \frac{2 - α}{1 - 0} = 2 \\ \Rightarrow & α & = \frac{5}{2} or & α = 0 . \end{aligned}$

Hence, the coordinates are $(0, 5 / 2)$ or $(0, 0)$ .

Straight Line and Pair of Straight Lines 1 Question 63

63. Two sides of a rhombus $A B C D$ are parallel to the lines $y = x + 2$ and $y = 7 x + 3$ . If the diagonals of the rhombus intersect at the point $(1, 2)$ and the vertex $A$ is on the $Y$ -axis, find possible coordinates of $A$ .

Solution:

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Straight Line and Pair of Straight Lines 1 Question 63

63. Two sides of a rhombus ABCD are parallel to the lines y=x+2 and y=7x+3. If the diagonals of the rhombus intersect at the point (1,2) and the vertex A is on the Y-axis, find possible coordinates of A.

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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63. Two sides of a rhombus $A B C D$ are parallel to the lines $y = x + 2$ and $y = 7 x + 3$ . If the diagonals of the rhombus intersect at the point $(1, 2)$ and the vertex $A$ is on the $Y$ -axis, find possible coordinates of $A$ .