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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.

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

The locus of the mid point of the portion intercepted between the axes by the line $x{\,}cos\alpha+y{\,}sin{\,} \alpha=p$, where $p\inR$, is

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

The line x cos(alpha) + y sin(alpha) = p intercepts the axes at (p / cos(alpha), 0) and (0, p / sin(alpha)). The midpoint (h, k) of these intercepts is h = p / (2 cos(alpha)) and k = p / (2 sin(alpha)). Squaring and adding 1/h^2 + 1/k^2 yields 4/p^2.

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

Locus of the midpoints of the intercepts between the co-ordinate Axes by the lines passing through (a, 0) does not intersect

  1. X axis

  2. Y axis

  3. Y=x

  4. Y=a
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Any line passing through (a, 0) has an intercept form x/p + y/q = 1 where it passes through (a, 0), meaning a/p + 0/q = 1, so p = a. The intercepts are (a, 0) and (0, q), so their midpoint has coordinates (h, k) = (a/2, q/2). This gives q = 2k, and since the y-intercept is q, the line intersects the y-axis at (0, 2k). However, the line cannot pass through the origin if it has distinct x and y intercepts, meaning q cannot be zero, which translates to the locus not intersecting the y-axis because h = a/2 is a vertical line parallel to the y-axis.