which line intersects the parabola (y = x^2 + 6x - 2) at two points?

a. (y = 6x - 4)

b. (y = 6x - 3)

c. (y = 6x - 2)

d. (y = 6x - 1)

Question

which line intersects the parabola (y = x^2 + 6x - 2) at two points?

a. (y = 6x - 4)

b. (y = 6x - 3)

c. (y = 6x - 2)

d. (y = 6x - 1)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Systems of Nonlinear Equations", "Discriminant of Quadratic", "Number of Real Solutions" ], "new_concepts": [], "current_concepts": [ "Systems of Nonlinear Equations", "Discriminant of Quadratic", "Number of Real Solutions" ] } ## Equate the parabola and the general line equation $$ \begin{aligned} &y = x^2 + 6x - 2 \ &y = 6x + k \ &x^2 + 6x - 2 = 6x + k \ &x^2 - (2 + k) = 0 \end{aligned} $$ ## Apply the discriminant condition for two intersection points $$ \begin{aligned} &a = 1,\quad b = 0,\quad c = -(2 + k) \ &D = b^2 - 4ac = 0^2 - 4(1)(-(2 + k)) = 4(2 + k) \ &D > 0 \implies 4(2 + k) > 0 \implies k > -2 \end{aligned} $$ ## Evaluate the given options for k > -2 $$ \begin{aligned} &\text{Option A: } k = -4 \implies -4 > -2 \quad (\text{False}) \ &\text{Option B: } k = -3 \implies -3 > -2 \quad (\text{False}) \ &\text{Option C: } k = -2 \implies -2 > -2 \quad (\text{False}) \ &\text{Option D: } k = -1 \implies -1 > -2 \quad (\text{True}) \end{aligned} $$

A. $y = 6x - 4$

B. $y = 6x - 3$

C. $y = 6x - 2$

D. $y = 6x - 1$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Systems of Nonlinear Equations" ] }

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QUESTION IMAGE

Question

Explanation:

Equate the parabola and the general line equation

Apply the discriminant condition for two intersection points

Evaluate the given options for k > -2

Answer: