Question

Condition for line 4x+3y+k=0 to intersect y2=8x is

Answer 1

$Let \: us \: find \: the \: point \: of \\ intersection \: of \: the \: line \: with \: the \\ curve. \\ for \: this \: substitute \: 4x = - 3y - k \\ in \: {y}^{2} = 8x \: and \: solve$

${y}^{2} = - 6y - 2k \\ {y}^{2} + 6y + 2k = 0 \\ this \: equation \: has \: real \: roots \: if \\ \: discriminant \: > 0\\ {6}^{2} - 4 \times 1 \times 2k > 0 \\ k < \frac{9}{2}$

$thus \: for \: k < \frac{9}{2} \: \: the \: line \: intersects \\ the \: curve$