College Algebra: Coordinate Geometry and Circles Study Guide

Q1. For the points P and Q, find (i) the distance d(P, Q) and (ii) the coordinates of the midpoint M of line segment PQ.

• (a) P(−5, −6), Q(7, −1)

• (b) P(8, 2), Q(3, 5)

• (c) P(−6, −5), Q(6, 10)

• (d) P(3\sqrt{2}, 4\sqrt{5}), Q(\sqrt{2}, -\sqrt{5})

Background

Topic: Distance and Midpoint Formulas in the Coordinate Plane

This question tests your ability to use the distance and midpoint formulas to analyze points in the plane.

Key formulas:

• Distance between two points $P(x_1, y_1)$ and $Q(x_2, y_2)$: $d(P, Q) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Midpoint of segment $PQ$: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

Step-by-Step Guidance

1. Identify the coordinates of $P$ and $Q$ for each part.

2. Apply the distance formula by substituting the $x$ and $y$ values into $d(P, Q)$.

3. Simplify the expressions inside the square root, but do not compute the final value yet.

4. For the midpoint, add the $x$-coordinates and divide by 2, and do the same for the $y$-coordinates.

5. Write the midpoint as an ordered pair, but do not simplify to decimals yet.

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Q2. Determine whether the three points are the vertices of a right triangle.

• (a) (−6, −4), (0, −2), (−10, 8)

• (b) (−4, 1), (1, 4), (−6, −1)

Background

Topic: Right Triangles in the Coordinate Plane

This question tests your ability to use the distance formula and the Pythagorean Theorem to determine if three points form a right triangle.

Key concepts:

• Distance formula (see above)

• Pythagorean Theorem: For a right triangle with sides $a$, $b$, and hypotenuse $c$, $a^2 + b^2 = c^2$

Step-by-Step Guidance

1. Label the points as $A$, $B$, and $C$.

2. Calculate the distances $AB$, $BC$, and $AC$ using the distance formula.

3. Identify the largest distance (potential hypotenuse).

4. Check if the sum of the squares of the two shorter sides equals the square of the longest side.

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Q3. Determine whether the three points are collinear.

• (a) (0, −7), (−3, 5), (2, −15)

• (b) (0, 9), (−3, −7), (2, 19)

Background

Topic: Collinearity of Points

This question tests your ability to determine if three points lie on the same straight line.

Key concepts:

• Points are collinear if the slopes between each pair are equal.

• Slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Step-by-Step Guidance

1. Label the points as $A$, $B$, and $C$.

2. Calculate the slope between $A$ and $B$.

3. Calculate the slope between $B$ and $C$.

4. Compare the two slopes to see if they are equal.

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Q4. If a vertical line is drawn through the point (4, 3), at what point will it intersect the x-axis?

Background

Topic: Equations of Vertical Lines and Intercepts

This question tests your understanding of vertical lines and how to find their intersection with the x-axis.

Key concepts:

• A vertical line has an equation $x = a$ for some constant $a$.

• The x-axis is where $y = 0$.

Step-by-Step Guidance

1. Write the equation of the vertical line passing through $(4, 3)$.

2. Set $y = 0$ to find the intersection with the x-axis.

3. Substitute $y = 0$ into the equation and solve for the corresponding $x$ value.

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Q5. If a horizontal line is drawn through the point (4, 3), at what point will it intersect the y-axis?

Background

Topic: Equations of Horizontal Lines and Intercepts

This question tests your understanding of horizontal lines and how to find their intersection with the y-axis.

Key concepts:

• A horizontal line has an equation $y = b$ for some constant $b$.

• The y-axis is where $x = 0$.

Step-by-Step Guidance

1. Write the equation of the horizontal line passing through $(4, 3)$.

2. Set $x = 0$ to find the intersection with the y-axis.

3. Substitute $x = 0$ into the equation and solve for the corresponding $y$ value.

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Q6. Decide whether or not each equation has a circle as its graph. If it does, give the center and radius. If it does not, describe the graph.

• (a) $x^2 + y^2 + 6x + 8y + 9 = 0$

• (b) $x^2 + y^2 -4x + 12y = -4$

• (c) $4x^2 + 4y^2 + 4x -16y -19 = 0$

• (d) $x^2 + y^2 + 2x -6y + 14 = 0$

• (e) $x^2 + y^2 -6x -6y + 18 = 0$

• (f) $9x^2 + 9y^2 -6x + 6y -23 = 0$

Background

Topic: Equations of Circles (Completing the Square)

This question tests your ability to recognize the general form of a circle and rewrite it in center-radius form by completing the square.

Key concepts and formulas:

• General form of a circle: $x^2 + y^2 + Dx + Ey + F = 0$

• Center-radius form: $(x - h)^2 + (y - k)^2 = r^2$

• Complete the square for $x$ and $y$ terms to rewrite the equation.

Step-by-Step Guidance

1. Group $x$ and $y$ terms together.

2. Complete the square for both $x$ and $y$ terms.

3. Rewrite the equation in the form $(x - h)^2 + (y - k)^2 = r^2$ if possible.

4. If the coefficients of $x^2$ and $y^2$ are not equal or not both positive, describe the graph accordingly.

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Q7. Find the equation of a circle with center at (−4, 3), passing through the point (5, 8). Write it in center-radius form.

Background

Topic: Equation of a Circle Given Center and a Point

This question tests your ability to write the equation of a circle when given its center and a point on the circle.

Key formula:

• Center-radius form: $(x - h)^2 + (y - k)^2 = r^2$

• Radius $r$ is the distance from the center to the given point.

Step-by-Step Guidance

1. Identify the center $(h, k) = (-4, 3)$ and the point $(x_1, y_1) = (5, 8)$.

2. Use the distance formula to find $r$: $r = \sqrt{(x_1 - h)^2 + (y_1 - k)^2}$

3. Write the equation in the form $(x + 4)^2 + (y - 3)^2 = r^2$.

4. Substitute the value of $r^2$ (do not compute the final value yet).

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Q8. Find all points (x, y) with x = y that are 4 units from (1, 3).

Background

Topic: Locus of Points Satisfying Distance and Linear Constraints

This question tests your ability to find points that satisfy both a distance condition and a linear relationship.

Key concepts and formulas:

• Distance formula: $\sqrt{(x - 1)^2 + (y - 3)^2} = 4$

• Constraint: $x = y$

Step-by-Step Guidance

1. Substitute $y = x$ into the distance equation.

2. Simplify the equation to involve only $x$.

3. Solve the resulting equation for $x$ (do not compute the final values yet).

4. Write the corresponding $(x, y)$ pairs.

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Q9. Find all points satisfying x + y = 0 that are 8 units from (−2, 3).

Background

Topic: Locus of Points Satisfying Distance and Linear Constraints

This question tests your ability to find points that satisfy both a distance condition and a linear relationship.

Key concepts and formulas:

• Distance formula: $\sqrt{(x + 2)^2 + (y - 3)^2} = 8$

• Constraint: $x + y = 0$

Step-by-Step Guidance

1. Express $y$ in terms of $x$ using the constraint ($y = -x$).

2. Substitute $y = -x$ into the distance equation.

3. Simplify the equation to involve only $x$.

4. Solve for $x$ (do not compute the final values yet), then find $y$.

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Q10. Find the coordinates of all points whose distance from (1, 0) is $\sqrt{10}$ and whose distance from (5, 4) is $\sqrt{10}$.

Background

Topic: Intersection of Circles (Systems of Equations)

This question tests your ability to solve a system of equations involving two circles with equal radii.

Key concepts and formulas:

• Distance from $(x, y)$ to $(1, 0)$: $\sqrt{(x - 1)^2 + (y - 0)^2} = \sqrt{10}$

• Distance from $(x, y)$ to $(5, 4)$: $\sqrt{(x - 5)^2 + (y - 4)^2} = \sqrt{10}$

Step-by-Step Guidance

1. Write the two distance equations as $(x - 1)^2 + y^2 = 10$ and $(x - 5)^2 + (y - 4)^2 = 10$.

2. Expand both equations.

3. Subtract one equation from the other to eliminate quadratic terms and obtain a linear equation.

4. Solve the linear equation for $x$ or $y$ in terms of the other variable.

5. Substitute back into one of the original equations to find the possible points (do not compute the final values yet).

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