Q47 / 90 · JEE Main 2022 (25 Jun Shift 2) · Mathematics

If the line $y = 4 + k x, k > 0$ , is the tangent to the parabola $y = x - x^{2}$ at the point $P$ and $V$ is the vertex of the parabola, then the slope of the line through $P$ and $V$ is

Solution

Plotting the given data we have,

Now by diagram, Slope of tangent at $P =$ Slope of line $AP$

${(\frac{d y}{d x})}_{(α, α - α^{2})} = 1 - 2 α = \frac{α - α^{2} - 4}{α}$

On solving we get $α = - 2 \Rightarrow P (- 2, - 6)$

So, slope of $P V = \frac{5}{2}$

Hence, the correct answer is option: 'C'.