One side of a rectangle lies along the...

MathematicsVarious Forms of Equations of a LineJEE Advanced 1978Easy

One side of a rectangle lies along the line $4 x + 7 y + 5 = 0$ . Two of its vertices are $(- 3, 1)$ and $(1, 1)$ . Find the equations of the other three sides.

Answer:4x + 7y - 11 = 0, 7x - 4y - 3 = 0, 7x - 4y + 25 = 0

Visualized Solution (English)

Analyze the Given Side and Vertices

Given side $L_{1}$ : $4 x + 7 y + 5 = 0$
Given vertices: $A (- 3, 1)$ and $C (1, 1)$

Verify the Second Vertex .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C (1, 1)

Check $C (1, 1)$ in $L_{1}$ : $4 (1) + 7 (1) + 5 = 16 \neq = 0$
Conclusion: $C (1, 1)$ is the opposite vertex to $A (- 3, 1)$

Find the Equation of the Opposite Side .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C D

Side $C D$ is parallel to $A B$ : $4 x + 7 y + k = 0$
Since $C D$ passes through $C (1, 1)$ :
$4 (1) + 7 (1) + k = 0 \Rightarrow k = - 11$
Equation of side $C D$ : $4 x + 7 y - 11 = 0$

Find the Equation of Adjacent Side .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } B C

Side $B C$ is perpendicular to $A B$ : $7 x - 4 y + k^{'} = 0$
Since $B C$ passes through $C (1, 1)$ :
$7 (1) - 4 (1) + k^{'} = 0 \Rightarrow k^{'} = - 3$
Equation of side $B C$ : $7 x - 4 y - 3 = 0$

Find the Equation of the Final Side .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A D

Side $A D$ is parallel to $B C$ : $7 x - 4 y + k^{''} = 0$
Since $A D$ passes through $A (- 3, 1)$ :
$7 (- 3) - 4 (1) + k^{''} = 0 \Rightarrow k^{''} = 25$
Equation of side $A D$ : $7 x - 4 y + 25 = 0$

Summary of Results

Key Takeaways:
Opposite sides are parallel: $m_{1} = m_{2}$
Adjacent sides are perpendicular: $m_{1} \cdot m_{2} = - 1$
Final Equations:
$4 x + 7 y - 11 = 0$
$7 x - 4 y - 3 = 0$
$7 x - 4 y + 25 = 0$

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