Straight Line and Pair of Straight Lines 1 Question 62
62. One of the diameters of the circle circumscribing the rectangle $A B C D 4 y=x+7$. If $A$ and $B$ are the points $(-3,4)$ and $(5,4)$ respectively, then find the area of rectangle.
(1985, 3M)
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Solution:
- Let $O$ be the centre of circle and $M$ be mid-point of $A B$.
Then, $\quad O M \perp A B \quad \Rightarrow \quad M(1,4)$
Since, slope of $A B=0$
Equation of straight line $M O$ is $x=1$ and equation of diameter is $4 y=x+7$.
$\Rightarrow \quad$ Centre is $(1,2)$. Also, $O$ is mid-point of $B D$
$\Rightarrow \frac{\alpha+5}{2}, \frac{\beta+4}{2}=(1,2) \Rightarrow \alpha=-3, \beta=0$
$\therefore \quad A D=\sqrt{(-3+3)^{2}+(4-0)^{2}}=4$
and $\quad A B=\sqrt{64+0}=8$
Thus, area of rectangle $=8 \times 4=32$ sq units
