For what value of k does the line 2y - x + k | vedclass

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(B) Step $1$: Solve the given equations simultaneously.Given that the line $2y - x + k = 0$ touches the parabola $x^2 + 4y = 0$.From the line equation

Vedclass · Accepted Answer

$-1/2$

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(B) Step $1$: Solve the given equations simultaneously.Given that the line $2y - x + k = 0$ touches the parabola $x^2 + 4y = 0$.From the line equation,$2y = x - k$,so $y = \frac{x - k}{2}$.Substituting $y = \frac{x - k}{2}$ into the parabola equation $x^2 + 4y = 0$,we get:$x^2 + 4\left(\frac{x - k}{2}\right) = 0$$x^2 + 2(x - k) = 0$$x^2 + 2x - 2k = 0$Step $2$: Use the condition for tangency.For the line to touch the parabola,the quadratic equation must have equal roots,meaning the discriminant $D = 0$.For $ax^2 + bx + c = 0$,$D = b^2 - 4ac$.Here,$a = 1, b = 2, c = -2k$.$D = (2)^2 - 4(1)(-2k) = 0$$4 + 8k = 0$$8k = -4$$k = -1/2$.

For what value of $k$ does the line $2y - x + k = 0$ touch the parabola $x^2 + 4y = 0$?

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