Question

The circumcentre of the triangle formed by the lines $y=2x$, $y=x$ and $y=3x+4$ is

• A. $(6,8)$
• B. $(6,–8)$
• C. $(3,4)$
• D. $(-3,–4)$

Answer 1

The correct option is B

$(6,–8)$

Explanation for the correct option:

Step 1: Equating circumradius to find the relation between $\left(x,y\right)$

The triangle is formed by lines $y=x$, $y=2x$ and $y=3x+4$.

Therefore, the coordinates of the triangle are $A(0,0),B(-4,-8)$ and $C(-2,-2)$.

Let the circumcentre of the triangle be $P(x,y)$.

Then, ${\mathrm{PA}}^2={\mathrm{PB}}^2$

$$\begin{gathered} \Rightarrow x^2+y^2={(x+4)}^2+{(y+8)}^2 \\ \Rightarrow x^2+y^2=x^2+8x+16+y^2+64+16y \\ \Rightarrow 8x+16y+80=0 \\ \Rightarrow x+2y+10=0\dots \dots \left(1\right) \end{gathered}$$

Step 2: Equate circumradius ${\left(PA\right)}^2={\left(PC\right)}^2$

Similarly

$$\begin{gathered} \Rightarrow x^2+y^2={(x+2)}^2+{(y+2)}^2 \\ \Rightarrow x^2+y^2=x^2+4x+4+y^2+4+4y \\ \Rightarrow 4x+4y+8=0 \\ \Rightarrow x+y+2=0\dots \dots \left(2\right) \end{gathered}$$

Step 3: Finding the circumcentre

Subtracting equation $\left(1\right)$ from equation $\left(2\right)$, we get

$y=-8$.

Substituting the value of $y$ in equation$\left(2\right)$ we get

$x=6$

Therefore, the circumcentre of the triangle is $(6,-8)$.

Hence, option (B) is the correct answer.