y axis divides the line segment joining the points A (- 4, 1) and B (1, 1) in the ratio 4 : 1
Step-by-step explanation:
Let assume that y axis divides the line segment joining the points A (- 4, 1) and B (1, 1) in the ratio k : 1 at (0, y).
We know,
Section Formula
If A(x₁, y₁) and B(x₂, y₂) be two points in the cartesian plane and C(x, y) be the point which divides AB internally in the ratio m₁ : m₂, then the coordinates of C is given by
Let A(x₁, y₁) and B(x₂, y₂) be two points in the cartesian plane and C(x, y) be the point which divides AB internally in the ratio m₁ : m₂, then the coordinates of C is given by
Centroid of a triangle is defined as the point at which the medians of the triangle meet and is represented by the symbol G.
Let A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) be the vertices of a triangle and G(x, y) be the centroid of the triangle, then the coordinates of G is given by
Answers & Comments
Answer:
y axis divides the line segment joining the points A (- 4, 1) and B (1, 1) in the ratio 4 : 1
Step-by-step explanation:
Let assume that y axis divides the line segment joining the points A (- 4, 1) and B (1, 1) in the ratio k : 1 at (0, y).
We know,
Section Formula
If A(x₁, y₁) and B(x₂, y₂) be two points in the cartesian plane and C(x, y) be the point which divides AB internally in the ratio m₁ : m₂, then the coordinates of C is given by
[tex]\begin{gathered} \boxed{\tt{ (x, y) = \bigg(\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}}\bigg)}} \\ \end{gathered} \\ [/tex]
So, on substituting the values in above formula, we get
[tex]\sf \: (0,y) = \bigg(\dfrac{1 \times k - 4 \times 1}{k + 1}, \: \dfrac{1 \times k + 1 \times 1}{k + 1} \bigg) \\ [/tex]
[tex]\sf \: (0,y) = \bigg(\dfrac{ k - 4}{k + 1}, \: \dfrac{k + 1}{k + 1} \bigg) \\ [/tex]
On comparing x- coordinate, we get
[tex]\sf \: \dfrac{ k - 4}{k + 1} = 0 \\ [/tex]
[tex]\sf \: { k - 4} = 0 [/tex]
[tex]\implies\sf \: k = 4\\ [/tex]
Hence, y axis divides the line segment joining the points A (- 4, 1) and B (1, 1) in the ratio 4 : 1
[tex]\rule{190pt}{2pt}[/tex]
[tex] {{ \sf{Additional\:Information}}}[/tex]
1. Distance Formula
Let A(x₁, y₁) and B(x₂, y₂) be two points in the cartesian plane, then distance between A and B is given by
[tex]\begin{gathered}\boxed{\tt{ AB \: = \sqrt{ {(x_{2} - x_{1}) }^{2} + {(y_{2} - y_{1})}^{2} }}} \\ \end{gathered} \\ [/tex]
2. Section formula
Let A(x₁, y₁) and B(x₂, y₂) be two points in the cartesian plane and C(x, y) be the point which divides AB internally in the ratio m₁ : m₂, then the coordinates of C is given by
[tex]\begin{gathered} \boxed{\tt{ (x, y) = \bigg(\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}}\bigg)}} \\ \end{gathered} \\ [/tex]
3. Mid-point formula
Let A(x₁, y₁) and B(x₂, y₂) be two points in the coordinate plane and C(x, y) be the mid-point of AB, then the coordinates of C is given by
[tex]\begin{gathered}\boxed{\tt{ (x,y) = \bigg(\dfrac{x_{1}+x_{2}}{2}, \dfrac{y_{1}+y_{2}}{2}\bigg)}} \\ \end{gathered} \\ [/tex]
4. Centroid of a triangle
Centroid of a triangle is defined as the point at which the medians of the triangle meet and is represented by the symbol G.
Let A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) be the vertices of a triangle and G(x, y) be the centroid of the triangle, then the coordinates of G is given by
[tex]\begin{gathered}\boxed{\tt{ (x, y) = \bigg(\dfrac{x_{1}+x_{2}+x_{3}}{3}, \dfrac{y_{1}+y_{2}+y_{3}}{3}\bigg)}} \\ \end{gathered} \\ [/tex]
5. Area of a triangle
Let A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) be the vertices of a triangle, then the area of triangle is given by
[tex]\begin{gathered}\boxed{\tt{ Area =\dfrac{1}{2}\bigg|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\bigg|}} \\ \end{gathered} \\ [/tex]