Tag: mid-point formula

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Questions Related to mid-point formula

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

If $(2, 3), (-4, 5), (1, -2)$ are the midpoints of the sides $\vec{BC}, \vec{CA}, \vec{AB}$ of $\triangle ABC$, then the equation of $\vec{AB}$ is 

  1. $3x-y-5=0$
  2. $x+3y+5=0$
  3. $x+3y-11=0$
  4. $3x-y+17=0$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let vertices be A(x1, y1), B(x2, y2), C(x3, y3). Using the given midpoints, set up midpoint equations to find the coordinates of vertices A and B. Once vertices A and B are determined, find the equation of the line passing through them, which results in x + 3y + 5 = 0.

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

The point on $X-axis$ equidistant from $(2,3)$and $(1,5)$ is

  1. $\left( \dfrac { -13 }{ 2 } ,0 \right) $
  2. $\left( \dfrac { 13 }{ 2 } ,0 \right) $
  3. $(13,0)$
  4. $none\ of\ these$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let the point be (x, 0). Equidistance means (x-2)^2 + (0-3)^2 = (x-1)^2 + (0-5)^2. Expanding: x^2 - 4x + 4 + 9 = x^2 - 2x + 1 + 25. Simplifying: -4x + 13 = -2x + 26, so -2x = 13, x = -13/2. The point is (-13/2, 0).

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

Let ${P} _{1}$ and ${P} _{2}$ be two fixed points in $xy-plane$. A line ${L} _{1}=0$ passes through ${P} _{1}$ intersects $y-axis$ at $B$ and the line ${L} _{2}=0$ passes through ${P} _{2}$ and intersects $x-axis$ at $A$. If ${L} _{1}=0$ and ${L} _{2}=0$ are perpendicular then the locus of mid-point of$AB$ is

  1. $Straight line$
  2. $Circle$
  3. $Ellipse$
  4. $Parabola$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The locus of the midpoint of a segment whose endpoints move on axes while the lines are perpendicular results in a straight line relationship based on the fixed points P1 and P2.

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

The point (5,0) on y-axis is equidistant from (-1,2) and (3,4).

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A point on the y-axis has coordinates (0, y), but the question states the point is (5,0), which actually lies on the x-axis, creating a contradiction in the premise. Furthermore, calculating the distance from (5,0) to (-1,2) gives sqrt((5+1)^2 + (0-2)^2) = sqrt(36 + 4) = sqrt(40), and to (3,4) gives sqrt((5-3)^2 + (0-4)^2) = sqrt(4 + 16) = sqrt(20). Since these distances are not equal, the statement is false.

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

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

  1. $\left( -6,0 \right )$
  2. $\left( -7,0 \right )$
  3. $\left( 7,0 \right )$
  4. $\left( -5,0 \right )$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let the point on the X-axis be (x, 0). Since it is equidistant from (3,5) and (4,2), equate the squares of their distances: (x - 3)^2 + (0 - 5)^2 = (x - 4)^2 + (0 - 2)^2. Expanding this gives x^2 - 6x + 9 + 25 = x^2 - 8x + 16 + 4, which simplifies to 2x = -14, so x = -7. Thus, the point is (-7, 0).

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

Let P be the point (1, 0) and Q a point on the curve ${ y }^{ 2 }=8x$. The locus of mid point of PQ is-

  1. ${ y }^{ 2 }-4x+2=0$
  2. ${ y }^{ 2 }+4x+2=0$
  3. ${ x }^{ 2 }+4y+2=0$
  4. ${ x }^{ 2 }-4y+2=0$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$P=(1,0), Q=(h,k)$


$k^2=8h$


Let $(\alpha, \beta)$ be the mid-point of PQ.

$\alpha =\dfrac{h+1}{2}, \beta =\dfrac{k+0}{2}$

$2\alpha-1=h, 2\beta=k$

$(2\beta)^2=8(2\alpha-1)$

$\beta^2=4\alpha-2$

$\implies y^2-4x+2=0$

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

The co -ordinates of the midpoint of a line segment joining $ p(5,7) $ and $ Q (-3,3) $ are........

  1. $ (2,4) $
  2. $ (1,5 ) $
  3. $ (4,2 ) $
  4. $ (2,5 ) $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Given,

$P(5,7),Q(-3,3)$

mid point is given by,

$(x,y)=\left ( \dfrac{5-3}{2},\dfrac{7+3}{2} \right )$

$\Rightarrow (x,y)=(1,5)$
Multiple choice maths constructions mid-point formula midpoints division of a line segment

If Q is a variable point on $x^2=4y$ and O is the origin, the locus of mid point OQ is equation of 

  1. an ellipse

  2. a parabola

  3. hyperbola

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

Let Q be (t^2, 2t) on the parabola x^2 = 4y. The midpoint OQ is (h, k) = (t^2 / 2, t / 2). Eliminating the parameter t gives t = 2k, so h = (2k)^2 / 2 = 2k^2, which rearranges to y^2 = (1/2)x, representing another parabola.