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JEE 2023 Mathematics PYQ — Let , and be the mid-points of the sides of a triangle with incen… | Mathem Solvex | Mathem Solvex

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JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

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 $(α + β 3, 0)$ , where $α$ and $β$ are rational numbers, then $\frac{α}{β ^{2}}$ is equal to

Choose the correct answer:

A.
6
B.
8
(Correct Answer)
C.
9
D.
12

Correct Answer:

Explanation

So, $D = (\frac{4}{2 + 2 + 2 2}, \frac{4}{2 + 2 + 2 2})$

$= (\frac{2}{2 + 2}, \frac{2}{2 + 2})$

$= (\frac{2}{2 + 2} \times \frac{2 - 2}{2 - 2}, \frac{2}{2 + 2} \times \frac{2 - 2}{2 - 2})$

$= (2 - 2, 2 - 2)$

$∴ y^{2} = 4 a x$

$(2 - 2)^{2} = 4 a (2 - 2)$

$\Rightarrow 4 a = 2 - 2 \Rightarrow a = \frac{2 - 2}{4}$

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

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

So,

$\frac{α}{β ^{2}} = \frac{1/2}{1/16} = 8$

Explanation

So, $D = (\frac{4}{2 + 2 + 2 2}, \frac{4}{2 + 2 + 2 2})$

$= (\frac{2}{2 + 2}, \frac{2}{2 + 2})$

$= (\frac{2}{2 + 2} \times \frac{2 - 2}{2 - 2}, \frac{2}{2 + 2} \times \frac{2 - 2}{2 - 2})$

$= (2 - 2, 2 - 2)$

$∴ y^{2} = 4 a x$

$(2 - 2)^{2} = 4 a (2 - 2)$

$\Rightarrow 4 a = 2 - 2 \Rightarrow a = \frac{2 - 2}{4}$

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

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

So,

$\frac{α}{β ^{2}} = \frac{1/2}{1/16} = 8$

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