Consider the region $R=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x \geq 0\right.$ and $\left.y^{2} \leq 4-x\right\}$. Let $\mathcal{F}$ be the family of all circles that are contained in $R$ and have centers on the $x$-axis. Let $C$ be the circle that has largest radius among the circles in $\mathcal{F}$. Let $(\alpha, \beta)$ be a point where the circle $C$ meets the curve $y^{2}=4-x$.
The radius of the circles $C$ is____.

Question

Consider the region $R=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x \geq 0\right.$ and $\left.y^{2} \leq 4-x\right\}$. Let $\mathcal{F}$ be the family of all circles that are contained in $R$ and have centers on the $x$-axis. Let $C$ be the circle that has largest radius among the circles in $\mathcal{F}$. Let $(\alpha, \beta)$ be a point where the circle $C$ meets the curve $y^{2}=4-x$.
The radius of the circles

C

is____.

Aicharya · Accepted Answer

equation of normal at P is y - &#946; = 2 &#946; x - 4 + &#946; 2 Passing through ( r , 0 ) &#8658; - 1 = 2 r - 4 + &#946; 2 &#160; . . . i or &#160;&#160; &#946; = 0 &#160; . . . i i Also r 2 = r + &#946; 2 - 4 2 + &#946; 2 &#160; . . . i i i from i and i i i r = 3 2 , - 5 2 so radius of circle is 1 . 5 and &#945; = 2 from i i and i i i r = 2 but in this case circle intersect at three points

Consider the region $R=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x \geq 0\right.$ and $\left.y^{2}…

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