Let A(0, 1), B(1, 1) and C(1, 0) be the mid-points of the sides of a triangle with incentre at the point D. If the focus of the parabola y 2 = 4 a x passing through D is ( 2 , 0 ) , where and are rational numbers, then 2 is equal to [2023

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Solution

Q.
Let A(0, 1), B(1, 1) and C(1, 0) be the mid-points of the sides of a triangle with incentre at the point D. If the focus of the parabola $y^{2} = 4 a x$ passing through D is $(α + β \sqrt{2}, 0)$ , where $α$ and $β$ are rational numbers, then $\frac{α}{β^{2}}$ is equal to [2023]

1 12

2 6

3 $\frac{9}{2}$

4 8

Ans.
(4)

$a = O P = 2, b = O Q = 2, c = P Q = 2 \sqrt{2}$

$Incentre is given by$

$D \equiv (\frac{a x_{1} + b x_{2} + c x_{3}}{a + b + c}, \frac{a y_{1} + b y_{2} + c y_{3}}{a + b + c})$

$D \equiv (\frac{4}{2 + 2 + 2 \sqrt{2}}, \frac{4}{2 + 2 + 2 \sqrt{2}}) \equiv (2 - \sqrt{2}, 2 - \sqrt{2})$

Now, $y^{2} = 4 a x$ passes through $D$ .

So, ${(2 - \sqrt{2})}^{2} = 4 \times a (2 - \sqrt{2}) \Rightarrow a = \frac{2 - \sqrt{2}}{4}$

The focus of parabola $y^{2} = 4 a x$ is $(a, 0)$

So, $(\frac{2 - \sqrt{2}}{4}, 0) \equiv (α + β \sqrt{2}, 0)$

$\Rightarrow α = \frac{2}{4} = \frac{1}{2}, β = - \frac{1}{4}$

$∴$ $\frac{α}{β^{2}} = 8$

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