1- the perpendicular bisector of line segment joining the points a(1,5) b(4,6) cuts y axis at which point 2-if the - Maths - Coordinate Geometry

Question

1- the perpendicular bisector of line segment joining the points
a(1,5) b(4,6) cuts y axis at which point
2-if the point p (2,1)lies on the line segment joining points
a(4,2 ) and b (8,4) then
a- ap=1/3ab , b- ap=pb, c- pb=1/3ab d- ap=1/2 ab choose the
correct option
i m getting the relation ab=4 ab am i right or wrong plz specify
this
3- show that a (-3,-1), b- (-4,-1), c(3,3) , d (4,3) r the
vertices of rhombus i m not getting rhombus

Lalit Mehra · Accepted Answer

1 The given point are A(1, 5) and B(4, 6) Let the line CD be the perpendicular bisector of line segment AB Suppose CD intersect AB in E âˆ´ Coordinates of E $= (\frac{4 + 1}{2}, \frac{6 + 5}{2}) = (\frac{5}{2}, \frac{11}{2})$ Let the slope of CD be m Since AB âŠ¥ CD, âˆ´ Slope of CD Ã— Slope of AB = â€“ 1 $\Rightarrow m \times (\frac{6 - 5}{4 - 1}) = - 1 \Rightarrow m \times (\frac{1}{3}) = - 1 \Rightarrow m = - 3$ âˆ´ Equation of CD is $y - \frac{11}{2} = - 3 (x - \frac{5}{2}) [Point slope form : y - y_{1} = m (x - x_{1})] ∴ 2 y - 11 = - 3 (2 x - 5) \Rightarrow 2 y - 11 = - 6 x + 15 \Rightarrow 6 x + 2 y - 26 = 0$ âˆ´ Equation of perpendicular bisector line segment AB is 6x + 2y â€“ 26 = 0 We know that, on the y-axis, x = 0 Putting x = 0 in 6x + 2y â€“ 26 = 0, we get 0 + 2y â€“ 26 = 0 âˆ´ 2y = 26 â‡’ y = 13 Thus, the perpendicular bisector of line segment AB cuts y-axis at (0, 13) 2 Given, P(2, 1) lies on the line segment joining A(4, 2) and B(8, 4) Using distance formula, $AP= \sqrt{(4 - 2)^{2} + (2 - 1)^{2}} = \sqrt{(2)^{2} + (1)^{2}} = \sqrt{4 + 1} = \sqrt{5} BP = \sqrt{(8 - 2)^{2} + (4 - 1)^{2}} = \sqrt{(6)^{2} + (3)^{2}} = \sqrt{36 + 9} = \sqrt{45} = 3 \sqrt{5} AB = \sqrt{(8 - 4)^{2} + (4 - 2)^{2}} = \sqrt{(4)^{2} + (2)^{2}} = \sqrt{16 + 4} = \sqrt{20} = 2 \sqrt{5} AB= 2 \sqrt{5} = 2 \times \sqrt{5} = 2 AP ∴ AP = \frac{1}{2} AP$ 3 The points A( â€“ 3, â€“ 1), B( â€“ 4, â€“ 1), C( 3, 3) and D( 4, 3) are not the vertices of the rhombus