The ellipse frac{x^2}{a^2} + frac{y^2}{b^2} = | vedclass

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(C) The condition for the line $y = mx + c$ to intersect the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ at real points is that the discriminant o

Vedclass · Accepted Answer

$a^2m^2 \ge c^2 - b^2$

Vedclass · Answer

(C) The condition for the line $y = mx + c$ to intersect the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ at real points is that the discriminant of the resulting quadratic equation must be non-negative,or equivalently,the line must not be strictly outside the ellipse.The condition for tangency is $c^2 = a^2m^2 + b^2$.For the line to intersect the ellipse at real points,the line must be a secant or tangent,which implies $c^2 \le a^2m^2 + b^2$.Rearranging this inequality,we get $a^2m^2 \ge c^2 - b^2$.

The ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and the straight line $y = mx + c$ intersect in real points only if

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