national senior certificate: mathematics: paper ii - supplementary
page 17 of 28

question 8

the diagram represents a model of a steel arch which is an arc of a circle centre c. the width, ab, of the arch is 16 metres and its maximum height, eo, is 2 metres.

one of the vertical pillars that supports the arch is at a point p, the midpoint of ao.

o (0; 0).

(a) determine the equation of the circle with centre c in the form:
$x^2 + y^2 + cx + dy + e = 0$.

Question

national senior certificate: mathematics: paper ii - supplementary
page 17 of 28

question 8

the diagram represents a model of a steel arch which is an arc of a circle centre c. the width, ab, of the arch is 16 metres and its maximum height, eo, is 2 metres.

one of the vertical pillars that supports the arch is at a point p, the midpoint of ao.

o (0; 0).

(a) determine the equation of the circle with centre c in the form:
$x^2 + y^2 + cx + dy + e = 0$.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [], "new_concepts": [ "Equation of a Circle", "Coordinate Geometry" ], "current_concepts": [ "Equation of a Circle", "Coordinate Geometry" ] } ## Determine coordinates of points on the circle $$ \begin{aligned} &O = (0, 0) \ &A = (-8, 0), \quad B = (8, 0) \ &E = (0, 2) \ &C = (0, y_c) \quad \text{where } y_c < 0 \end{aligned} $$ ## Calculate the radius and center of the circle $$ \begin{aligned} &R^2 = (x - x_c)^2 + (y - y_c)^2 \ &R^2 = (0 - 0)^2 + (2 - y_c)^2 = (2 - y_c)^2 \ &R^2 = (8 - 0)^2 + (0 - y_c)^2 = 64 + y_c^2 \ &(2 - y_c)^2 = 64 + y_c^2 \ &4 - 4y_c + y_c^2 = 64 + y_c^2 \ &-4y_c = 60 \ &y_c = -15 \ &R^2 = 64 + (-15)^2 = 289 \end{aligned} $$ ## Expand the standard equation of the circle $$ \begin{aligned} &(x - 0)^2 + (y - (-15))^2 = 289 \ &x^2 + (y + 15)^2 = 289 \ &x^2 + y^2 + 30y + 225 - 289 = 0 \ &x^2 + y^2 + 30y - 64 = 0 \end{aligned} $$ $$x^2 + y^2 + 30y - 64 = 0$$ { "elements": [ { "type": "point", "params": [[0, 0]], "properties": { "name": "O(0,0)", "size": 3, "color": "#583C87", "withLabel": true } }, { "type": "point", "params": [[-8, 0]], "properties": { "name": "A(-8,0)", "size": 3, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[8, 0]], "properties": { "name": "B(8,0)", "size": 3, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[0, 2]], "properties": { "name": "E(0,2)", "size": 3, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[0, -15]], "properties": { "name": "C(0,-15)", "size": 3, "color": "#F2557F", "withLabel": true } }, { "type": "circle", "params": [[0, -15], 17], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2, "dash": 2 } }, { "type": "line", "params": [[-10, 0], [10, 0]], "properties": { "strokeColor": "#cccccc", "strokeWidth": 1 } }, { "type": "line", "params": [[0, -20], [0, 5]], "properties": { "strokeColor": "#cccccc", "strokeWidth": 1 } } ], "timestamps": [0.5] } { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Equation of a Circle" ] }

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Question

Explanation:

Determine coordinates of points on the circle

Calculate the radius and center of the circle

Expand the standard equation of the circle

Answer: