Tangent Values E6953A

1. **State the problem:** We have a line with equation $$y = -5x + k + 5$$ and a curve with equation $$y = 7 - kx - x^2$$. The line is tangent to the curve, meaning they touch at exactly one point. We need to find: (a) The two possible values of $k$ for which the line is tangent to the curve. (b) For each $k$, find the equation of the tangent, the equation of the curve, and the coordinates of the point of contact. --- 2. **Use the tangent condition:** Since the line is tangent to the curve, their equations are equal at exactly one $x$ value. Set the line equal to the curve: $$-5x + k + 5 = 7 - kx - x^2$$ Rearranging all terms to one side: $$-5x + k + 5 - 7 + kx + x^2 = 0$$ Simplify constants: $$x^2 + (k - 5)x + (k - 2) = 0$$ This quadratic in $x$ must have exactly one solution (tangency), so its discriminant $D$ is zero: $$D = b^2 - 4ac = (k - 5)^2 - 4 \times 1 \times (k - 2) = 0$$ --- 3. **Solve for $k$:** Expand and simplify: $$(k - 5)^2 - 4(k - 2) = 0$$ $$k^2 - 10k + 25 - 4k + 8 = 0$$ $$k^2 - 14k + 33 = 0$$ Solve quadratic for $k$: $$k = \frac{14 \pm \sqrt{14^2 - 4 \times 1 \times 33}}{2} = \frac{14 \pm \sqrt{196 - 132}}{2} = \frac{14 \pm \sqrt{64}}{2}$$ $$k = \frac{14 \pm 8}{2}$$ Two values: $$k_1 = \frac{14 + 8}{2} = 11$$ $$k_2 = \frac{14 - 8}{2} = 3$$ --- 4. **Find tangent points for each $k$:** Recall quadratic in $x$: $$x^2 + (k - 5)x + (k - 2) = 0$$ For tangency, one root: $$x = \frac{-b}{2a} = \frac{-(k - 5)}{2} = \frac{5 - k}{2}$$ Calculate $x$ for each $k$: - For $k=11$: $$x = \frac{5 - 11}{2} = \frac{-6}{2} = -3$$ - For $k=3$: $$x = \frac{5 - 3}{2} = \frac{2}{2} = 1$$ --- 5. **Find $y$ coordinates of contact points:** Use line equation $y = -5x + k + 5$: - For $k=11$, $x=-3$: $$y = -5(-3) + 11 + 5 = 15 + 16 = 31$$ - For $k=3$, $x=1$: $$y = -5(1) + 3 + 5 = -5 + 8 = 3$$ --- 6. **Summarize answers:** (a) The two possible values of $k$ are: $$k = 11 \quad \text{and} \quad k = 3$$ (b) For each $k$: - When $k=11$: - Tangent line: $$y = -5x + 11 + 5 = -5x + 16$$ - Curve: $$y = 7 - 11x - x^2$$ - Point of contact: $$(-3, 31)$$ - When $k=3$: - Tangent line: $$y = -5x + 3 + 5 = -5x + 8$$ - Curve: $$y = 7 - 3x - x^2$$ - Point of contact: $$(1, 3)$$