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

Question

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]

Smart Achievers · Accepted Answer

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

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

$∴$ $\frac{α}{β^{2}} = 8$

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

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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 y2=4ax passing through D is (α+β2,0), where α and β are rational numbers, then αβ2 is equal to [2023]

1 12

2 6

3 92

4 8

Ans. (4) a=OP=2, b=OQ=2, c=PQ=22 Incentre is given by D≡(ax1+bx2+cx3a+b+c,ay1+by2+cy3a+b+c) D≡(42+2+22, 42+2+22)≡(2-2, 2-2) Now, y2=4ax passes through D. So, (2-2)2=4×a(2-2)⇒a=2-24 The focus of parabola y2=4ax is (a,0) So, (2-24,0)≡(α+β2, 0) ⇒ α=24=12, β=-14 ∴ αβ2=8

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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]

3 $\frac{9}{2}$