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Distance between two points

$d = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}$

Problems:

1. Find the distance between the points P(0, 0) and Q(

s i n θ, c o s θ

) Ans: 1

2. Find the distance between the points (1, 1) and (

s i n θ + 1, c o s θ + 1

) Ans: 1

3. The point A(m², 9) is equidistant from both the axes. Find the value of m. Ans:

\pm 3

4. Distance between origin and the point (p, 4) is 5 units. Find the value of p where p > 0. Ans: 3

5. Show that the points A(-5, 0), B(5, 0), C(5, 5), and D(-5, 5) are the four vertices of a rectangle. Hint: opposite sides are equal, angles are right angle.

6. P(x, y) is equidistant from the points A(4, 3) and C(-3, - 2). Show that 7x + 5y = 6. [C.B. 2023]

7. From the point P(x, y), the distance of y-axis is equal to the distance of P from the point D(4, 3). Show that, y² -6y - 8x + 25 = 0. [Dhaka Board 2021]

Midpoint

M = (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})

Problems:

1. The vertices of a triangle ABC are A(2, -1), B(-4, 2), and C(2, 5). Find the length of the median AD. Ans:

\frac{3}{2} \sqrt{13}

2. PQRS is a parallelogram where P(8, 3), Q(3, 8), R(-2, 3). Find the coordinates of the point S. Ans: (3, -2) Hint: diagonals of a parallelogram bisect each other.

Slop of a line

m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}

Problems:

1. Slope of a line is -4 and value of y-intercept is 2. Write the equation of the line.

2. If AB || CD for the points A(x, y), B(1, 2), C(2, 1), and D(-x, -y); prove that x - y + 1 = 0 [M.B. 2023]

3. If P(m²+2, 3m), Q(4, 8), and R(7, 5) points are colinear, find the value of m. [C.B. 2021]

4. Show that, the connecting straight line between points (3, -5) and (6, 4) form an acute angle with the positive side of x-axis. [Rajshahi Board 2023]

Slope of a line perpendicular to the with slope m is:

$- \frac{1}{m}$ ; negative reciprocal of the original line's slope.

Equation of a line through a point (x₁, y₁):

$y - y_{1} = m (x - x_{1})$

Problems:

1. Find of equation of a line with gradient -2 that passes through the point (-5, 5).

2. The straight line y = -3x + 2 passes through the point P(t, 8) and intersects at A and B on x-axis and y-axis respectively. Determine the value of t. Then find out the equation of straight line whose gradient is 3 and passes through P. [Dinajpur Board 2022]

Find intersetion point from given equations:

1. Determine the area of the triangle formed by three equation: y = x + 7, y = -x + 5, and y = 3. [Sylhet Board 2022]

2. Determine the ratio of the area of the triangles formed by the lines L₁ : 3x + 8y = 25 and L₂ : 9x + 2y = 31 with axes. [Cumilla Board 2022, Jashore Board 2022]

3. A(-4, 4), B(6, 4), C(6, -7), and D(4, -7) are four vertices of a quadrilateral. Determine the area of that portion of quadrilateral ABCD which resides in the fourth quadrant. [Rajshahi Board 2023]

406,

1. A is the point (-1, -5) and B is the point (3, 3).

Find the equation of the line perpendicular to AB which passes throught the midpoint of AB.

[ 4024/22/M/J/21 ]

2. The line L crosses the graph of y = 1 +

\frac{2}{x}

at x = 2 and x = 5

Find the equation of L.

[ 4024/22/M/J/19 ]

3. A line with gradient

- \frac{1}{3}

crosses the graph of

y = 1 + \frac{2}{x}

when x = 1 and when x = k.

Find k.

[ 4024/22/M/J/19 ]

4. Line L is perpendicular to the line 2y = x + 4.

Line L passes through the point (1, 8).

Show that the equation of line L is y = 10 - 2x

5. Use an algebraic method to find the coordinates of the point of intersection of the lines 2y = x + 4 and y = 23 - 2x.

6. A is the point (-1, 3) and B is the point (5, 5).

(i) Calculate the length AB.

(ii) Find the equation of the line perpendicular to AB that passes through the midpoint of AB.

[ 4024/21/M/J/18 ]

7. (a) The points (4, -3) and (0, 5) lie on the line L.

Find the equation of line L.

(b) The line M is parallel to line L and passes through the point (-2, 3).

Find the equation of line M.

[ 4024/21/O/N/17 ]

1. P is the point (3, -3) and Q is the point (1,5)

(a) Calculate the length of PQ.

(b) Find the equation of the perpendicular bisector of PQ. [ 4024/21/M/J/22 ]

2. D is a point (4, 6) and E is the point (e, e).

(a) The length of DE is

\sqrt{20}

Form an equation in e and solve it to find the possible coordinates of E.

Show your working.

(b) F is the point (-f, 5f).

The gradient of the perpendicular bisector of DF is

\frac{3}{2}

(i) Find the value of f.

(ii) The equation of the perpendicular bisector of DF is 2y = 3x + k .

Find the value of k . [ 4024/22/M/J/22 ]

3. A is the point (-2, 3) and B is the point (4, 5).

(a) Find the coordinates of the midpoint of AB.

(b) Show that the equation of line AB is 3y = x + 11.

4024/21/O/N/23

1. a) The equation of line L is 4y = x - 5 .

i) Find the gradient of line L.

ii) Find the coordinate of the point where line L crosses the y-axis.

b) A is the point (4, 5) and B is the point (-2, 8).

i) Find the length of line AB.

ii) Find the equation of line AB. Give your answer in the form y = mx + c .

4024/22/O/N/23

1. The equation of line P is y = 4x - 3 .

Line L is perpendicular to line P .

Line L passes through the point (6, 4).

Find the coordinates of the point where line L crosses the x-axis.

4024/21/O/N/22

P is the point (r, 4) and Q is the point (t, u).

The midpoint of line PQ is (1, 3).

The gradient of line PQ is

- \frac{1}{4}

Find the value of each of r, t and u.

Coordinate Geometry