Question

line joining the points A(4 , -5) and B(4 , 5) divided by the point P such that AP/PB = 2/5 , Find the co-ordinates of P​

Answer 1

Answer:

The co-ordinates of P are $\left(4, \dfrac{-15}{7}\right)$.

Given:

• A(4,-5)

• B(4,5)

• AP:PB=2:5

To Calculate:

Co-ordinates of P

Explanation:

Let the co-ordinates of point P be P(a, b).

We know that, if a point divides a line in ratio $m_1:m_2$. Then the co-ordinates of the point that divides the line is $\\\left(\dfrac{m_1 x_2 + m_2 x_1}{m_1 + m_2} , \dfrac{m_1 y_2 + m_2 y_1}{m_1 + m_2}\right)$.

In the question:

• $x_1$ = 4
• $y_1$ = -5
• $x_2$ = 4
• $y_2$ = 5
• $m_1$ = 2
• $m_2$ = 5

Substituting the given values, we get:

P(a, b) = $\left(\dfrac{2\times4+5\times4}{2+5},\dfrac{2\times5+5\times-5}{2+5}\right)$

P(a, b) = $\left(\dfrac{28}{7},\dfrac{-15}{7}\right)$

P(a, b) = $\left(4,\dfrac{-15}{7}\right)$

Therefore, the co-ordinates of P are P($\left(4,\dfrac{-15}{7}\right)$).

Other Formulas:

1) Slope of Line

• Slope of a non-vertical line passing through points A($x_1,y_1$) and B($x_1,y_2$) is:

$\:\:\:\:\:\:\:\:\:\:\:\:m=\dfrac{y_2-y_1}{x_2-x_1}$

• If a line makes an angle $\theta$ with the positive side of x-axis, then the slope of line is:

$\:\:\:\:\:\:\:\:\:\:\:\:m = \tan\theta$

2) Equation of a Line

• Equation of a line parallel to x-axis at a distance b is:

$\:\:\:\:\:\:\:\:\:\:\:\:y = b$(where b is constant)

• Equation of a line parallel to y-axis at a distance is:

$\:\:\:\:\:\:\:\:\:\:\:\:x = a$(where a is constant)

• Equation of a line having a slope and making an intercept with y-axis is:

$\:\:\:\:\:\:\:\:\:\:\:\:y = mx+c$(where m is the slope and c is the y-intercept made by line)

• Equation of a line when the line is passing through one point and slope is given:

$\:\:\:\:\:\:\:\:\:\:\:\:(y-y_1)=m(x-x_1)$ (where $x_1,y_1$ are co-ordinates of point through which line passes and m is the slope.

• Equation of a non-vertical line passing through two points is:

$\:\:\:\:\:\:\:\:\:\:\:\:(y-y_1)=\dfrac{y_2-y_1}{x_2-x_1}(x-x_1)$ (where $(x_1,y_1)\:and\:(y_1,y_2)$ are co-ordinates of two points through which line passes.

Conditions for two lines to be:

• Parallel is that the slope of both lines should ve equal.

$\:\:\:\:\:\:\:\:\:\:\:\:$Let the slope of first line and second line be $m_1$ and $m_2$ respectively.

Therefore, the two lines are parallel if $m_1=m_2$

• Perpendicular is that the product of the slopes of the two lines should be equal to -1.

$\:\:\:\:\:\:\:\:\:\:\:\:$ Let the slope of first and second line be $m_1$ and $m_2$ respectively.

Therefore, the two lines are perpendicular if $m_1\times m_2=-1$.