### Straight Lines Question 3

#### Question 3 - 25 January - Shift 2

The equations of two sides of a variable triangle are $x=0$ and $y=3$, and its third side is a tangent to the parabola $y^{2}=6 x$. The locus of its circumcentre is :

(1) $4 y^{2}-18 y-3 x-18=0$

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

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

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

## Show Answer

#### Answer: (3)

#### Solution:

#### Formula: Mid Point Formula, Slope intercept form

$y^{2}=6 x \quad \text{and} \quad y^{2}=4 ax$

$\Rightarrow 4 a=6 \Rightarrow a=\frac{3}{2}$

$y=m x+\frac{3}{2 m} ;(m \neq 0)$

$h=\frac{6 m-3}{4 m^{2}}, k=\frac{6 m+3}{4 m}$, Now eliminating $m$ and we get

$\Rightarrow 3 h=2(-2 k^{2}+9 k-9)$

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

$\therefore $ Option (3) is correct.