If the line 2x - 3y = k is tangent to the par | vedclass

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(D) Given the line equation is $2x - 3y = k$,which can be written as $x = \frac{3y + k}{2}$.Substituting this into the parabola equation $y^2 = 6x$:$y

Vedclass · Accepted Answer

$-27/4$

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(D) Given the line equation is $2x - 3y = k$,which can be written as $x = \frac{3y + k}{2}$.Substituting this into the parabola equation $y^2 = 6x$:$y^2 = 6 \left( \frac{3y + k}{2} \right)$$y^2 = 3(3y + k)$$y^2 - 9y - 3k = 0$.Since the line is tangent to the parabola,the quadratic equation must have equal roots,meaning the discriminant $D = 0$.$D = (-9)^2 - 4(1)(-3k) = 0$$81 + 12k = 0$$12k = -81$$k = -\frac{81}{12} = -\frac{27}{4}$.

If the line $2x - 3y = k$ is tangent to the parabola $y^2 = 6x$,then what is the value of $k$?

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