Straight Line and Pair of Straight Lines 1 Question 67
67. The points $(1,3)$ and $(5,1)$ are two opposite vertices of a rectangle. The other two vertices lie on the line $y=2 x+c$. Find $c$ and the remaining vertices. (1981,4M)
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Answer:
Correct Answer: 67. $(-4,-7)$
Solution:
- Since, diagonals of rectangle bisect each other, so mid point of $(1,3)$ and $(5,1)$ must satisfy $y=2 x+c$, i.e. $(3,2)$ lies on it.
$\Rightarrow \quad c=-4$
$\therefore$ Other two vertices lies on $y=2 x-4$
Let the coordinate of $B$ be $(x, 2 x-4)$.
$\therefore$ Slope of $A B \cdot$ Slope of $B C=-1$
$$ \begin{aligned} & \Rightarrow \quad \frac{2 x-4-3}{x-1} \cdot \frac{2 x-4-1}{x-5}=-1 \\ & \Rightarrow \quad\left(x^{2}-6 x+8\right)=0 \\ & \Rightarrow \quad x=4,2 \\ & \Rightarrow \quad y=4,0 \end{aligned} $$
Hence, required points are $(4,4),(2,0)$.
