The line 3x + 4y – 24 = 0 intersects the x-axis at A and y-axis at B. Then the circumcentre of the triangle OAB where O is the origin is

Concept:

Circumcentre: circumcenter is the point where the perpendicular bisectors of a triangle intersect.

The circumcenter of a right triangle lies exactly at the midpoint of the hypotenuse (longest side).

Calculation:

Given:

3x + 4y – 24 = 0

⇒ 3x + 4y = 24

\(\Rightarrow \frac{{3{\rm{x}}}}{{24}} + {\rm{\;}}\frac{{4{\rm{y}}}}{{24}} = 1 \Rightarrow \frac{{\rm{x}}}{8} + {\rm{\;}}\frac{{\rm{y}}}{6} = 1\)

Its x-intercept A = (8, 0) and y-intercept B = (0, 6)

Since, AOB is a right angled triangle. So, the mid-point of hypotenuse is the circumcentre.

Hence, circumcentre = \(\left( {\frac{{8\; + \;0}}{2},{\rm{\;}}\frac{{0\; + \;6}}{2}} \right)\)

∴ circumcentre = (4, 3)