Tag: division of a line segment

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Multiple choice maths constructions mid-point formula midpoints division of a line segment

The locus of the mid-point of that chord of parabola which subtends right angle on the vertex will be

  1. $y ^ { 2 } - 2 a x + 8 a ^ { 2 } = 0$
  2. $y ^ { 2 } = a ( x - 4 a )$
  3. $y ^ { 2 } = 4 a ( x - 4 a )$
  4. $y ^ { 2 } + 3 a x + 4 a ^ { 2 } = 0$
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Explanation

For a parabola y^2 = 4ax, the chord subtending a right angle at the vertex has the equation y = mx + 2am. The midpoint (h, k) of this chord satisfies k = mh + 2am and the property that the chord is y = (2a/k)x - 4a^2/k. Substituting and simplifying leads to the locus y^2 = 2a(x - 4a), which is y^2 - 2ax + 8a^2 = 0.

Multiple choice maths constructions mid-point formula midpoints division of a line segment

Find a point on the y-axis which equidistant from the points $A(6,5)$ and $B(-4,3)$

  1. $(0,9)$
  2. $(9,0)$
  3. $(3,0)$
  4. $(4,0)$
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Explanation

Let the point be (0, y). Equidistance means the square of the distances to (6,5) and (-4,3) are equal: (0-6)^2 + (y-5)^2 = (0-(-4))^2 + (y-3)^2. This simplifies to 36 + y^2 - 10y + 25 = 16 + y^2 - 6y + 9, which results in 61 - 10y = 25 - 6y, or 4y = 36, so y = 9.

Multiple choice maths constructions mid-point formula midpoints division of a line segment

If $(-6,-4)$ and $(3,5)$ are the extremities of the diagonals of a parallelogram and $(-2,1)$ is its third vertex, then its fourth vertex is 

  1. $(-1,0)$
  2. $(0,-1)$
  3. $(-1,1)$

  4. none of these
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Explanation
Given,

$P(-6,-4),Q(3,5),R(-2,1),S(\alpha ,\beta )$

Let $P$ and $Q$ are the extremities of diagonals of a parallelogram, and 

$R$ and $S$ will be the extremities of diagonals of a parallelogram

Now,

midpoint of $PQ=\dfrac{3-6}{2},\dfrac{5-4}{2}=\dfrac{-3}{2},\dfrac{1}{2}$

midpoint of $RS\Rightarrow \dfrac{-2+\alpha }{2}=-\dfrac{3}{2}$

$\Rightarrow \alpha =-3+2=-1$

Now,

$\dfrac{\alpha +\beta }{2}=\dfrac{1}{2}$

$\Rightarrow \beta =0$

Therefore, coordinates of 4th vertex is $(-1,0)$
Multiple choice maths constructions mid-point formula midpoints division of a line segment

A (a,b) and (0,0) are two fixed points, ${ M } _{ 1 }$ is the mid points of AB, ${ M } _{ 2 }$ is the midpoint of $A{ M } _{ 1 },{ M } _{ 3 }$ is the midpoint of $A{ M } _{ 2 }$ and so on then ${ M } _{ 5 }$ =in

  1. $\left( \dfrac { 7a }{ 8 } ,\dfrac { 7b }{ 8 } \right) $
  2. $\left( \dfrac { 15a }{ 16 } ,\dfrac { 15b }{ 16 } \right) $
  3. $\left( \dfrac { 31a }{ 32 } ,\dfrac { 15b }{ 32 } \right) $
  4. $\left( \dfrac { 63a }{ 64 } ,\dfrac { 15b }{ 64 } \right) $
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Multiple choice maths constructions mid-point formula midpoints division of a line segment

If an triangle ABC, A = {1, 10}, circumference = $\left( -\dfrac { 1 }{ 3 } ,\dfrac { 2 }{ 3 }  \right) $ and orthocenter = $\left( \dfrac { 11 }{ 3 } ,\dfrac { 4 }{ 3 }  \right) $ then the co-ordinate of mid-point of side opposite to A is ________.

  1. (1, 11/3)

  2. (1, 5)

  3. (1, -3)

  4. (1, 6)
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