A rectangle R with end points of one of its sides as (1,2) and (3,6) is inscribed in a cir…

Question

A rectangle $R$ with end points of one of its sides as $(1,2)$ and $(3,6)$ is inscribed in a circle. If the equation of a diameter of the circle is $2x-y+4=0$, then the area of $R$ is:

Answer: 16

Accepted Answer

Side length $a = \sqrt{(3-1)^2+(6-2)^2}=\sqrt{4+16}=\sqrt{20}$.

Slope of this side $=2$, matching the diameter's slope, so the diameter is parallel to this side and hence perpendicular to the adjacent side $b$; the perpendicular distance from the diameter (which passes through the rectangle's center) to this side equals $b/2$.

Midpoint of the given side: $(2,4)$. Distance from $(2,4)$ to $2x-y+4=0$:
$$\frac{|4-4+4|}{\sqrt5} = \frac4{\sqrt5}$$

So $\dfrac b2 = \dfrac4{\sqrt5} \implies b=\dfrac8{\sqrt5}$.

$$\text{Area} = ab = \sqrt{20}\times\frac8{\sqrt5} = 2\sqrt5\times\frac8{\sqrt5}=16$$

Solutions · 1