Coordinate Problems Involving Circles and Lines

15.6: Coordinate Problems Involving Circles and Lines. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 6 of 63 parts Practice Flashcards

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Horizontal line intersections

Objective: Substitute a fixed y-value into a circle equation.

Prerequisite skills: circle equations, line equations, and right triangles. A horizontal line y=c fixes one coordinate. Substitute it into the circle equation and solve for x. A positive remaining square gives two intersections, zero one, and a negative value none. Common error: using the full radius as the horizontal difference when the line is not through the center. Restrictions and interpretation: Intersection coordinates must satisfy both equations; a tangent corresponds to exactly one shared point.

Formulas

$(x−h)²=r²−(c−k)²$

Worked examples

Try itThe circle x²+y²=12100 meets the line y=66. What is the greater intersection x-coordinate?Show answer

Substitute y=66: x²=12100−4356=7744. Thus x=±88; the greater is 88. Therefore the requested answer is 88.

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Horizontal line intersections

Objective: Substitute a fixed y-value into a circle equation.

Formulas

$(x−h)²=r²−(c−k)²$

Worked examples

Try itThe circle x²+y²=12100 meets the line y=66. What is the greater intersection x-coordinate?Show answer

Substitute y=66: x²=12100−4356=7744. Thus x=±88; the greater is 88. Therefore the requested answer is 88.

Tangency by center distance

Objective: Compare line-to-center distance with radius.

For a horizontal line y=c and center (h,k), the perpendicular distance is |c−k|. Equality with r gives tangency; smaller gives two intersections; larger gives none. A vertical line uses |c−h|. Common error: using a signed distance instead of an absolute value. Restrictions and interpretation: Intersection coordinates must satisfy both equations; a tangent corresponds to exactly one shared point.

Formulas

horizontal line distance=|c−k|

Worked examples

Try itA circle has center (4,24) and radius 27. What is the y-coordinate of its upper horizontal tangent?Show answer

The upper tangent lies exactly one radius above the center. Its y-value is 24+27=51. The center-to-line distance is 27, giving one tangency point. Therefore the requested answer is 51.

Line through a centered circle

Objective: Use substitution for a nonhorizontal line.

Substitute a line’s y-expression into the circle equation, expand, and solve the resulting quadratic. Pair each x with the y from that same line. A line through the center of a nondegenerate circle meets it at opposite points. Common error: solving the circle and line separately without matching a common point. Restrictions and interpretation: Intersection coordinates must satisfy both equations; a tangent corresponds to exactly one shared point. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

$x²+(mx)²=r² ⇒ (1+m²)x²=r²$

Worked examples

Try itThe circle x²+y²=15625 intersects the line y=(4/3)x. What is the positive x-coordinate of an intersection?Show answer

Substitute y=(4/3)x: x²+(16/9)x²=15625. Thus (25/9)x²=15625, so x²=5625. The positive root is x=75; then y=100, satisfying both equations. Therefore the requested answer is 75.

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