A parabola y = ax^2 + bx + c crosses the x-ax | vedclass

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(D) Let the points of intersection of the parabola with the $x$-axis be $A(\alpha, 0)$ and $B(\beta, 0)$.Since the parabola $y = ax^2 + bx + c$ passes

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$\sqrt{\frac{c}{a}}$

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(D) Let the points of intersection of the parabola with the $x$-axis be $A(\alpha, 0)$ and $B(\beta, 0)$.Since the parabola $y = ax^2 + bx + c$ passes through these points,$\alpha$ and $\beta$ are the roots of the equation $ax^2 + bx + c = 0$.From the properties of roots,the product of the roots is $\alpha \beta = \frac{c}{a}$.Let the circle pass through $A$ and $B$. The origin $O(0, 0)$ lies on the $x$-axis.The power of the point $O$ with respect to the circle is given by $OT^2 = OA \cdot OB$,where $T$ is the point of tangency.Since $A$ and $B$ are at distances $\alpha$ and $\beta$ from the origin respectively,$OA = \alpha$ and $OB = \beta$.Therefore,$OT^2 = \alpha \beta = \frac{c}{a}$.Thus,the length of the tangent $OT = \sqrt{\frac{c}{a}}$.

$A$ parabola $y = ax^2 + bx + c$ crosses the $x$-axis at $(\alpha, 0)$ and $(\beta, 0)$,both to the right of the origin. $A$ circle also passes through these two points. The length of a tangent from the origin to the circle is:

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