The equations of two sides of a variable triangle are x = 0 and y =...

Given,

Equation of parabola is ${y}^{2} = 6x$,

Also the general equation of a parabola is ${y}^{2} = 4ax$

On comparing, we get

$⇒ 4a = 6 ⇒ a = \dfrac{3}{2}$

Tangent on parabola ${y}^{2} = 4ax$ is given by $y = mx + \dfrac{a}{m}$,

So, $y = mx + \dfrac{3}{2m}; ( m \ne 0 )$

Now using the midpoint formula we get,

$h = \dfrac{6m - 3}{4{m}^{2}}, k = \dfrac{6m + 3}{4m}$

Now, eliminating $m$ and we get $3h = 2\left(- 2{k}^{2} + 9k - 9\right)$

$⇒ 3h = - 4{k}^{2} + 18k - 18$

$⇒ 4{k}^{2} - 18k + 3h + 18 = 0$

$∴$ The locus of its circumcentre is

$4{y}^{2} - 18y + 3x + 18 = 0$