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Advanced Maths: Short Tricks on Coordinate Geometry pls tell

3 years ago

Important Short Tricks on Coordinate Geometry

Equation of line parallel to y-axis

X = a

For Example: A Student plotted four points on a graph. Find out which point represents the line parallel to y-axis.

a) (3,5)

b) (0,6)

c) (8,0)

d) (-2, -4)

Solution: Option (C)

Equation of line parallel to x-axis

Y = b

For Example: A Student plotted four points on a graph. Find out which point represents the line parallel to x-axis.

a) (3,5)

b) (0,6)

c) (8,0)

d) (-2, -4)

Solution: Option (B)

Equations of line

a) Normal equation of line

ax + by + c = 0

b) Slope – Intercept Form

y = mx + c Where, m = slope of the line & c = intercept on y-axis

For Example: What is the slope of the line formed by the equation 5y - 3x - 10 = 0?

Solution: 5y - 3x - 10 = 0, 5y = 3x + 10

Y = 3/5 x + 2

Therefore, slope of the line is = 3/5

c) Intercept Form

x/A + y/B = 1, Where, A & B are x-intercept & y-intercept respectively

For Example: Find the area of the triangle formed the line 4x + 3 y – 12 = 0, x-axis and y-axis?

Solution: Area of triangle is = ½ * x-intercept * y-intercept.

Equation of line is 4x + 3 y – 12 = 0

4x + 3y = 12,

4x/12 + 3y/12 = 1

x/3 + y/4 = 1

Therefore area of triangle = ½ * 3 * 4 = 6

d) Trigonometric form of equation of line, ax + by + c = 0

x cos θ + y sin θ = p,

Where, cos θ = -a/ √(a2 + b2) , sin θ = -b/ √(a2 + b2) & p = c/√(a2 + b2)

e) Equation of line passing through point (x1,y1) & has a slope m

y - y1 = m (x-x1)

Slope of line =

3 years ago

Normal equation of line

ax + by + c = 0

b) Slope – Intercept Form

y = mx + c Where, m = slope of the line & c = intercept on y-axis

For Example: What is the slope of the line formed by the equation 5y - 3x - 10 = 0?

Solution: 5y - 3x - 10 = 0, 5y = 3x + 10

Y = 3/5 x + 2

Therefore, slope of the line is = 3/5

c) Intercept Form

x/A + y/B = 1, Where, A & B are x-intercept & y-intercept respectively

For Example: Find the area of the triangle formed the line 4x + 3 y – 12 = 0, x-axis and y-axis?

Solution: Area of triangle is = ½ * x-intercept * y-intercept.

Equation of line is 4x + 3 y – 12 = 0

4x + 3y = 12,

4x/12 + 3y/12 = 1

x/3 + y/4 = 1

Therefore area of triangle = ½ * 3 * 4 = 6

d) Trigonometric form of equation of line, ax + by + c = 0

x cos θ + y sin θ = p,

Where, cos θ = -a/ √(a2 + b2) , sin θ = -b/ √(a2 + b2) & p = c/√(a2 + b2)

e) Equation of line passing through point (x_{1},y_{1}) & has a slope m

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

- Slope of line = y
_{2}- y_{1}/x_{2}- x_{1 }= - coefficient of x/coefficient of y

- Angle between two lines

Tan θ = ± (m_{2} – m_{1})/(1+ m_{1}m_{2}) where, m_{1} , m_{2} = slope of the lines

Note: If lines are parallel, then tan θ = 0

If lines are perpendicular, then cot θ = 0

For Example: If 7x - 4y = 0 and 3x - 11y + 5 = 0 are equation of two lines. Find the acute angle between the lines?

Solution: First we need to find the slope of both the lines.

7x - 4y = 0

⇒ y = 74x

Therefore, the slope of the line 7x - 4y = 0 is 74

Similarly, 3x - 11y + 5 = 0

⇒ y = 311x + 511

Therefore, the slope of the line 3x - 11y + 5 = 0 is = 311

Now, let the angle between the given lines 7x - 4y = 0 and 3x - 11y + 5 = 0 is θ

Now,

Tan θ = ± (m_{2} – m_{1})/(1+ m_{1}m_{2}) = ±[(7/4)−(3/11)]/[1+(7/4)*(3/11)] = ± 1

Since θ is acute, hence we take, tan θ = 1 = tan 45°

Therefore, θ = 45°

Therefore, the required acute angle between the given lines is 45°.

- Equation of two lines parallel to each other

ax + by + c_{1} = 0

ax + by + c_{2} = 0

Note: Here, coefficient of x & y are same.

- Equation of two lines perpendicular to each other

ax + by + c_{1} = 0

bx - ay + c_{2} = 0

Note: Here, coefficient of x & y are opposite & in one equation there is negative sign.

- Distance between two points (x
_{1}, y_{1}), (x_{2}, y_{2})

D = √ (x_{2} – x_{1})^{2} + (y_{2} – y_{1})^{2}

For Example: Find the distance between (-1, 1) and (3, 4).

Solution: D = √ (x_{2} – x_{1})^{2} + (y_{2} – y_{1})^{2}

= √ (3 – (-1))^{2} + (4 – 1)^{2} = √(16 + 9) = √25 = 5

- The midpoint of the line formed by (x
_{1}, y_{1}), (x_{2}, y_{2})

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

- Area of triangle whose coordinates are (x
_{1}, y_{1}), (x_{2}, y_{2}), (x_{3}, y_{3})

½ I x_{1} (y_{2} – y_{3}) + x_{2} (y_{3} – y_{1}) + x_{3}(y_{1} – y_{2}) I

For Example: Find area of triangle whose vertices are (1, 1), (2, 3) and (4, 5).

Solution: We have (x_{1}, y_{1}) = (1, 1), (x_{2}, y_{2}) = (2, 3) and (x_{3}, y_{3}) = (4, 5)

Area of Triangle = ½ I x_{1} (y_{2} – y_{3}) + x_{2} (y_{3} – y_{1}) + x_{3} (y_{1} – y_{2}) I

=1/2 I (1(3−5) +2(5−1) + 4(1−3)) I

=1/2 I(−2+8−8) =1/2 (−2) I = I−1I = 1

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