at how many points does the line with equation $y = -\frac{3}{4}x + \frac{25}{4}$ intersect the circle shown?

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there is not enough information to determine the number of points of intersection.

Question

at how many points does the line with equation $y = -\frac{3}{4}x + \frac{25}{4}$ intersect the circle shown?

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there is not enough information to determine the number of points of intersection.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [], "new_concepts": [ "Circle Equations", "Line-Circle Intersection", "Distance from Point to Line" ], "current_concepts": [ "Circle Equations", "Line-Circle Intersection", "Distance from Point to Line" ] } ## Find the equation of the circle $$ \begin{aligned} &\text{Center: } (h, k) = (0, 0) \ &\text{Radius: } r = 5 \ &x^2 + y^2 = 25 \end{aligned} $$ ## Calculate the distance from the center to the line $$ \begin{aligned} &y = -\frac{3}{4}x + \frac{25}{4} \implies 3x + 4y - 25 = 0 \ &d = \frac{|3(0) + 4(0) - 25|}{\sqrt{3^2 + 4^2}} \ &d = \frac{|-25|}{\sqrt{25}} = \frac{25}{5} = 5 \end{aligned} $$ ## Determine the number of intersection points $$ \begin{aligned} &d = r = 5 \ &\text{The line is tangent to the circle.} \ &\text{Number of intersection points} = 1 \end{aligned} $$

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There is not enough information to determine the number of points of intersection.

{ "elements": [ { "type": "circle", "params": [[0, 0], [5, 0]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2, "name": "x^2 + y^2 = 25", "withLabel": false } }, { "type": "functiongraph", "params": [{"js": "-0.75*x + 6.25", "latex": "y = -\frac{3}{4}x + \frac{25}{4}"}, -10, 10], "properties": { "strokeColor": "#F2557F", "strokeWidth": 2 } }, { "type": "point", "params": [[3, 4]], "properties": { "name": "(3, 4)", "size": 4, "color": "#8C55F2", "withLabel": true } } ], "timestamps": [0.5] } { "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Line-Circle Intersection" ] }

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QUESTION IMAGE

Question

Explanation:

Find the equation of the circle

Calculate the distance from the center to the line

Determine the number of intersection points

Answer: