Name: The Line Family
Uploaded: 2019-07-19
Duration: 4 min 33 s
Description: A family of lines is a set of lines that have something in common with each other. Learn more about the family of lines by watching this video.

Question

Find the position of the center of the circle circumscribing the triangle whose vertices are the points

(2, 3), (3, 4)

and

(6, 8)

.

Accepted Answer

Let the triangle be $A B C$ having coordinates $A (2, 3), B (3, 4)$ and $C (6, 8)$ and $O$ be center of circle.

Equation of the line $B C$ is given by

$\frac{y - 8}{4 - 8} = \frac{x - 6}{3 - 6}$

$\Rightarrow \frac{y - 8}{- 4} = \frac{x - 6}{- 3}$

$\Rightarrow 3 y - 24 = 4 x - 24$

$\Rightarrow 4 x - 3 y = 0$

Slope of $B C = \frac{4}{3}$

so, slope of line perpendicular to $B C = \frac{- 3}{4}$

Let $D$ be the mid- point of $B C$

Coordinates of $D$ are $(\frac{3 + 6}{2}, \frac{4 + 8}{2}) = (\frac{9}{2}, 6)$

Equation of line $O D$ which passes through $D (\frac{9}{2}, 6)$ and having slope $(\frac{- 3}{4})$ is

$(y - 6) = \frac{- 3}{4} (x - \frac{9}{2})$

$\Rightarrow 4 y - 24 = \frac{27}{2} - 3 x$

$\Rightarrow 8 y - 48 = 27 - 6 x$

$\Rightarrow 6 x + 8 y - 75 = 0 . . . (1)$

Similarly, equation of line $A B$ is

$\frac{y - 3}{4 - 3} = \frac{x - 2}{3 - 2}$

$\Rightarrow y - 3 = x - 2$

$\Rightarrow x - y + 1 = 0$

Slope of $A B = 1$

Slope of line perpendicular to $A B = - 1$

Coordinates of midpoint of $A B$ is

$E = (\frac{2 + 3}{2}, \frac{3 + 4}{2}) = (\frac{5}{2}, \frac{7}{2})$

Equation of line $O E$ is

$(y - \frac{y}{2}) = - 1 (x - \frac{5}{2})$

$\Rightarrow x + y - 6 = 0 . . . (2)$

Solving equations $(1)$ and $(2)$ , we get

$x = \frac{- 27}{2}, y = \frac{39}{2}$

Hence, coordinates of center $O$ is

$(\frac{- 27}{2}, \frac{39}{2})$

Since, the triangle is obtuse angled triangle $(∵ A B^{2} + B C^{2} < C A^{2})$ , the center of circumcircle lies outside the triangle.

Find the position of the center of the circle circumscribing the triangle whose vertices are the points $(2, 3), (3, 4)$ and $(6, 8)$ .

Important Questions on The Straight Line (Continued)

Learn and Prepare for any exam you want !

Find the position of the center of the circle circumscribing the triangle whose vertices are the points 2, 3, 3, 4 and 6, 8.

Important Questions on The Straight Line (Continued)

Learn and Prepare for any exam you want !

Find the position of the center of the circle circumscribing the triangle whose vertices are the points $(2, 3), (3, 4)$ and $(6, 8)$ .