Question

AMC12B Let P be the parabola with equation y=ax2+bx+c, and let l be the line y=mx+n. There are real numbers a, b, c, m, and n such that the line through (0,0) with slope m does not intersect P if and only if −2

a) 1/2
​b) −1/2
c) −1/4
d) 3/4

​

Answer 1

The value of
`\(a\)` that satisfies the condition where the line through
`\((0,0)\)` with slope
`\(m\)` does not intersect the parabola
`\(y=ax^2+bx+c\)` if and only if
`\(a = (1)/(2)\) (option a)` .

Step-by-step explanation:

To find the condition where the line
`\(y=mx+n\)` doesn't intersect the parabola
`\(y=ax^2+bx+c\)` when passing through
`\((0,0)\)` with slope
`\(m\)` , we consider the discriminant of the quadratic equation formed by their intersection.

Given the parabola
`\(y=ax^2+bx+c\)` and the line
`\(y=mx+n\)` , the point of intersection will satisfy
`\(ax^2+bx+c = mx+n\)` . For
`\((0,0)\)` to lie on the line,
`\(n\) must be \(0\)` . So, the equation becomes
`\(ax^2+bx = mx\)` .

For the line to not intersect the parabola, the discriminant of this quadratic equation
`(\(b^2-4ac\))` must be negative. Here,
`\(a\)` is the coefficient of
`\(x^2\), \(b\)` is the coefficient of
`\(x\), and \(c=0\)` .

Solving for the discriminant, we have
`\(b^2-4ac = (b)^2 - 4(a)(0)\)` , simplifying to
`\(b^2\)` . Since
`\(b\)` represents the coefficient of
`\(x\)` in the equation
`\(ax^2+bx = mx\)`, the condition for no intersection is
`\(b^2 < 0\)` .

This condition holds true only if
`\(b = 0\)` , implying
`\(a = (1)/(2)\)` to satisfy
`\(b^2 < 0\)` . Therefore,
`\(a = (1)/(2)\)` is the value for which the line with slope
`\(m\)` passing through
`\((0,0)\)` does not intersect the parabola.