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

The locus of the middle points of chords of length $4$ on the circle $x^ {2}+y^ {2}=16$

  1. A straight line

  2. A circle of radius
  3. A circle of radius $2\sqrt {3}$

  4. An ellipse
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For a circle of radius R = 4, the distance d of a chord of length L = 4 from the center is given by d = sqrt(R^2 - (L/2)^2). Here R = 4 and L/2 = 2, so d = sqrt(16 - 4) = sqrt(12) = 2(sqrt(3)). The locus of the midpoints of chords of constant length is a concentric circle whose radius is equal to this perpendicular distance from the center, which is 2(sqrt(3)).

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. $(4,0)$
Reveal answer Fill a bubble to check yourself
A Correct answer
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
Reveal answer Fill a bubble to check yourself
A Correct answer
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) $
Reveal answer Fill a bubble to check yourself
A Correct answer
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)
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice maths constructions mid-point formula midpoints division of a line segment

The point on X-axis which is equidistant from the point (3, 5) and (4, 2)

  1. (-6, 0)

  2. (-7, 0)

  3. (7, 0)

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let the point on x-axis equidistant from (3,5) and (4,2) be (x,0),


Then, Distance of point from (3,5) = DIstance of point from (4,2)
$\rightarrow \sqrt{(3-x)^2+(5-0)^2} = \sqrt{(4-x)^2+(2-0)^2}$
$\Rightarrow 9+x^2-6x+25=16+x^2-8x+4$
$\rightarrow 2x = -14$
$\rightarrow x = -7$
$\rightarrow $ Point is $(-7,0)$

Thus, B is the correct answer.

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

If $A(a, b)$ and $B(0, 0)$ are two fixed points. $M _1$ is the mid point of $\overline{AB}$, $M _2$ is the mid point of $\overline{AM _1}$, $M _3$ is the mid point of $\overline{AM _2}$ and so on, then $M _5$ is?

  1. $\left(\dfrac{7a}{8}, \dfrac{7b}{8}\right)$
  2. $\left(\dfrac{15a}{16}, \dfrac{15b}{16}\right)$
  3. $\left(\dfrac{31a}{32}, \dfrac{31b}{32}\right)$
  4. $\left(\dfrac{63a}{64}, \dfrac{63b}{64}\right)$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let A be (a, b) and B be (0, 0). The midpoint formula shows that each successive midpoint moves closer to A by a factor of half the remaining distance. Specifically, M1 = (a/2, b/2), M2 = a - (a - M1)/2, and following the geometric progression for n steps, the coordinate for Mn is given by (a(1 - 1/2^n), b(1 - 1/2^n)). For M5, substituting n = 5 gives 1 - 1/32 = 31/32, resulting in (31a/32, 31b/32).