Analytical Geometry Formulas Overview

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Questions and Answers

What is the gradient of a horizontal line?

  • 0 (correct)
  • 1
  • Undefined
  • Negative

Vertical lines have a gradient that is defined.

False (B)

What is the formula to find the distance between two points A(Ax, Ay) and B(Bx, By)?

d = √((Bx - Ax)² + (By - Ay)²)

The equation of a circle centered at the origin is given by __________.

<p>x² + y² = r²</p> Signup and view all the answers

Match the type of line to its gradient:

<p>Horizontal Line = 0 Vertical Line = Undefined Positive Line = m &gt; 0 Negative Line = m &lt; 0</p> Signup and view all the answers

What is the formula to determine the midpoint between points A(Ax, Ay) and B(Bx, By)?

<p>((Ax + Bx)/2, (Ay + By)/2) (B)</p> Signup and view all the answers

Collinear points have equal gradients between them.

<p>True (A)</p> Signup and view all the answers

What does the 'm' represent in the context of lines?

<p>Gradient or slope</p> Signup and view all the answers

What is the radius of the circle with the center at the origin?

<p>7 (C)</p> Signup and view all the answers

The distance between points C(0, 3) and D(2, 19) is longer than 15.

<p>True (A)</p> Signup and view all the answers

What is the midpoint of the diameter CD where C is (0, 3) and D is (2, 19)?

<p>(1, 11)</p> Signup and view all the answers

The equation of a circle in standard form is (x - ______)^2 + (y - ______)^2 = r^2.

<p>h, k</p> Signup and view all the answers

Match the following points with their characteristics:

<p>(2, 0) = x-intercept (0, 3) = y-intercept (1, 1) = neither (5, 0) = x-intercept</p> Signup and view all the answers

If the radius of a circle is 5 and it intersects the x-axis at points (2, 0) and (8, 0), what is the center of the circle?

<p>(5, 0) (B)</p> Signup and view all the answers

Which of the following can be used to find the center and radius of a circle from its equation?

<p>Standard form of a circle equation (D)</p> Signup and view all the answers

A circle can be defined by its center and diameter.

<p>True (A)</p> Signup and view all the answers

What equation represents a circle passing through the point (1, 5) and having its center at (3, 3)?

<p>(x - 3)^2 + (y - 3)^2 = 5</p> Signup and view all the answers

The equation of a circle always requires completing the square to extract its center and radius.

<p>True (A)</p> Signup and view all the answers

What is the key operation used to simplify the equation of a circle?

<p>Completing the square</p> Signup and view all the answers

The center of a circle in the form $(x - a)^2 + (y - b)^2 = r^2$ is located at the point (___, ___).

<p>a, b</p> Signup and view all the answers

Match the following components related to circles with their definitions:

<p>Center = The fixed point at the center of the circle Radius = The distance from the center to any point on the circle Circle Equation = Mathematical representation of a circle Diameter = The length of a line segment that passes through the center and connects two points on the circle</p> Signup and view all the answers

What is the correct formula for the circumference of a circle?

<p>$C = \pi d$ (A), $C = 2\pi r$ (C)</p> Signup and view all the answers

The center of the circle is located at the origin (0,0) if not specified.

<p>False (B)</p> Signup and view all the answers

What does 'sub in' mean in mathematical terms?

<p>Substitute values into an equation</p> Signup and view all the answers

The standard form of the equation of a circle is $(x - ______)^2 + (y - ______)^2 = r^2$.

<p>h, k</p> Signup and view all the answers

Match the following elements with their respective descriptions:

<p>r = Radius of the circle C = Circumference of the circle d = Diameter of the circle (h, k) = Center of the circle</p> Signup and view all the answers

Which of the following points would lie on a circle with a radius of 6 and center (0, 0)?

<p>(0, 6) (B), (6, 0) (D)</p> Signup and view all the answers

The equation $x^2 + y^2 = 49$ describes a circle with a radius of 7.

<p>True (A)</p> Signup and view all the answers

What do you obtain when you complete the square for the equation $x^2 + 11x + y^2 = 36$?

<p>A circle equation in standard form</p> Signup and view all the answers

Flashcards

Gradient

The slope of a line. It represents the steepness of a line and the direction it goes in. It is calculated by dividing the difference in y-coordinates by the difference in x-coordinates of two points on the line.

Point of intersection

The point where two lines intersect. This is the point where both lines share the same x-coordinate and y-coordinate.

Distance between two points

The distance between two points. This is the length of the line connecting the two points.

Diameter

The line that divides a circle into two equal halves. It passes through the center of the circle.

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Radius

The line from the center of a circle to any point on the circle's edge.

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Midpoint

The point that divides a line segment into two equal parts. The mid-point lies exactly halfway between the two endpoints.

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Trapezium

A shape with four sides, where one pair of opposite sides are parallel and equal in length, while the other pair of opposite sides are also parallel but not necessarily equal.

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Diagonal

A line segment connecting two opposite vertices of a trapezium.

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Substitution in equations

A process of substituting known values into a formula to solve for an unknown variable.

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Solving equations

A method of solving equations that involves performing operations on both sides of the equation to isolate the unknown variable.

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Equation of a circle

A method of finding the equation of a circle when given its center and radius.

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Simplifying expressions

The process of manipulating an expression to simplify it or to make it easier to solve.

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Expanding and simplifying

A technique used to rewrite expressions by multiplying out brackets and combining like terms.

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Y-intercept

The point where a graph crosses the y-axis.

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Inverse operations

A method for solving equations that involves isolating the unknown variable by performing inverse operations on both sides of the equation.

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Finding the equation of a circle

A method of finding the equation of a circle when given certain points on the circle and its center.

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Completing the Square (CTS)

A method used to rewrite an equation in the form (x-a)² + (y-b)² = r² to identify the centre (a, b) and radius (r) of a circle.

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Centre of a Circle: x-coordinate

In the form (x-a)² + (y-b)² = r², 'a' represents the x-coordinate of the centre of the circle.

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Centre of a Circle: y-coordinate

In the form (x-a)² + (y-b)² = r², 'b' represents the y-coordinate of the centre of the circle.

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Radius of a Circle

In the form (x-a)² + (y-b)² = r², 'r' represents the radius of the circle.

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Center of a Circle from Diameter

To find the center of a circle given its diameter, find the midpoint of the diameter. The midpoint will be the center of the circle.

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Finding the Equation of a Circle with x-intercepts

When finding the equation of a circle given x-intercepts, remember that the x-intercepts are points on the circle. These points can be used to find the radius or other information needed to complete the equation.

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Finding the Equation of a Circle through Two Points

To find the equation of a circle passing through two given points, substitute the coordinates of both points into the standard equation of a circle. This will give you two equations with two unknowns (h and k, the center of the circle). Solve for h and k.

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Radius in Circle Equation

When finding the equation of a circle, the radius (r) is a key component. Use the distance formula to find the radius if you know the center point and a point on the circle.

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Equation of a Circle: Summary

To find the equation of a circle when given key information like the center, radius, or points on the circle, use the standard equation of a circle: (x - h)^2 + (y - k)^2 = r^2. Substitute values and solve for missing components.

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Study Notes

Analytical Geometry Formulas

  • Straight Lines:
    • y = mx + c (where c is the y-intercept)
    • y - y₁ = m(x - x₁) (where (x₁, y₁) is a point on the line)
  • Point of Intersection: Use simultaneous equations.
  • Horizontal Line: y = c
  • Vertical Line: x = k
  • Distance Formula: AB = √((x₂ - x₁)² + (y₂ - y₁)²) (calculates distance between points A(x₁, y₁) and B(x₂, y₂))
  • Gradient (m): m = (y₂ - y₁) / (x₂ - x₁) (for points A(x₁, y₁) and B(x₂, y₂))
    • The gradient of a horizontal line is 0.
    • The gradient of a vertical line is undefined.

Angle of Inclination

  • To determine the angle of inclination, use: m = tan θ (θ is the angle of inclination)
  • Four things to remember about gradient: -If two lines are parallel, their gradients are equal (m₁=m₂). -If two lines are perpendicular, the product of their gradients is -1 (m₁ x m₂= -1). -Points are collinear if the gradient between any two points is the same. -Gradient is the tangent of the angle of inclination.

Midpoint Formula

  • The midpoint between points A(x₁, y₁) and B(x₂, y₂) is given by M = ((x₁ + x₂)/2, (y₁ + y₂)/2).

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Analytical Geometry Notes (PDF)

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