Sectional Formula MCQ Quiz - Objective Question with Answer for Sectional Formula - Download Free PDF

Calculation

Let the points be 4(-2, 4, 7) and B(3, -5, 8) on YZ-plane, x-coordinate = 0.

Let the ratio be K ∶ 1.

The coordinates of C

are \(\left(\frac{3 K-2}{K+1}, \frac{-5 K+4}{K+1}, \frac{8 K+7}{K+1}\right)\)

Clearly \(\frac{3 K-2}{K+1}=0 \Rightarrow 3 K=2 \Rightarrow K=\frac{2}{3}\)

Hence required ratio is 2 ∶ 3.

Hence option 2 is correct

Concept Used

Section Formula (Internal Division):

If a point P(x, y, z) divides the line segment joining A(x₁, y₁, z₁) and B(x₂, y₂, z₂) in the ratio m:n internally, then the coordinates of P are given by:

x = \(\frac{mx_2 + nx_1}{m+n}\)

y = \(\frac{my_2 + ny_1}{m+n}\)

z = \(\frac{mz_2 + nz_1}{m+n}\)

Calculation

Given;

Points: A(2, -1, 3) and B(4, 3, 1)

Ratio: m:n = 3:4

x = \(\frac{3(4) + 4(2)}{3+4}\) = \(\frac{12 + 8}{7}\) = \(\frac{20}{7}\)

y = \(\frac{3(3) + 4(-1)}{3+4}\) = \(\frac{9 - 4}{7}\) = \(\frac{5}{7}\)

z = \(\frac{3(1) + 4(3)}{3+4}\) = \(\frac{3 + 12}{7}\) = \(\frac{15}{7}\)

The coordinates of the point P are (\(\frac{20}{7}\), \(\frac{5}{7}\), \(\frac{15}{7}\)).

Hence option 3 is correct.

Answer : 3

Solution :

The given line segment has endpoints (3, 2, -1) and (6, 2, -2). We can find the coordinates of point P by using the section formula, which is applied here in its simplest form since P is somewhere on the line segment directly between the two given points.

Let the coordinates of P be (x, y, z). Since the z-coordinate of P is given as 5, and we know that Plies on the line segment, we can use the formula for a point dividing a line segment in a given ratio (in this case, since the z -coordinates are increasing from 3 to 6, and 5 lies two-thirds of the way from 3 to 6, the division will be in the ratio 1 ∶ 2).

The formula for a point P dividing the line segment with endpoints (x₁, y₁, z₁) and (x₂, y₂, z₂) in the ratio m : n is:

\(x=\frac{m x_{2}+n x_{1}}{m+n}\)

\(y=\frac{m y_{2}+n y_{1}}{m+n}\)

\(z=\frac{m z_{2}+n z_{1}}{m+n}\)

Substituting the given values, with m = 1 and n = 2 (as derived from the distances for the z-coordinates, where 5 is 2 units from 3 and 1 unit from 6), the endpoints are (3, 2, -1) and (6, 2, -2), respectively. Therefore, the y- coordinate of P would be calculated as:

\(y=\frac{1 \times 2+2 \times 2}{1+2}=\frac{2+4}{3}=\frac{6}{3}=2\)

Since the calculation for y straightforwardly results in 2, and the y-coordinates of both endpoints of the segment are 2, the y-coordinate of P remains constant throughout the line segment at 2. Thus, the correct answer is:

Option 3 : 2

Calculation

Let the point be \(G(h,k')\). Then the co-ordinates of the point will be given by section formula.

\(G(h,k')=\dfrac{2(1)+3(2)}{3+2},\dfrac{2(1)+3(4)}{3+2}\)

\(\Rightarrow\dfrac{8}{5},\dfrac{14}{5}\)

Since the line passes through G, we get

⇒ \(2(\dfrac{8}{5})+\dfrac{14}{5}=k\)

⇒ \(k=\dfrac{16+14}{5}\)

⇒ \(k=\dfrac{30}{5}\)

⇒ \(k=6\)

Hence option 3 is correct

Calculation

Let R be the point where the YZ-plane divides the line segment PQ in the ratio 2:3. The coordinates of R can be found using the section formula:

\(R = \left(\frac{2(3) + 3(\alpha)}{2+3}, \frac{2(\beta) + 3(4)}{2+3}, \frac{2(8) + 3(7)}{2+3}\right)\)

Since R lies on the YZ-plane, its x-coordinate is 0. Thus,

\(\frac{6 + 3\alpha}{5} = 0 \Rightarrow 6 + 3\alpha = 0 \Rightarrow \alpha = -2\)

Let S be the point where the ZX-plane divides the line segment PQ in the ratio 4:5. The coordinates of S can be found using the section formula:

\(S = \left(\frac{4(3) + 5(\alpha)}{4+5}, \frac{4(\beta) + 5(4)}{4+5}, \frac{4(8) + 5(7)}{4+5}\right)\)

Since S lies on the ZX-plane, its y-coordinate is 0. Thus,

\(\frac{4\beta + 20}{9} = 0 \Rightarrow 4\beta + 20 = 0 \Rightarrow \beta = -5\)

\(PQ = \sqrt{(3 - (-2))^2 + (-5 - 4)^2 + (8 - 7)^2}\)

\(PQ = \sqrt{(5)^2 + (-9)^2 + (1)^2}\)

\(PQ = \sqrt{25 + 81 + 1}\)

\(PQ = \sqrt{107}\)

Hence option 1 is correct

CONCEPT:

Let A (x1, y1) and B (x2, y2) be the two given points and the point P (x, y) divide the line joining the points A and B in the ratio m : n, then

• Point of internal division is given as: \(\left( {x,\;y} \right) = \left( {\frac{{m{x_2} + n{x_1}}}{{m + n}},\frac{{m{y_2} + n{y_1}}}{{m + n}}} \right)\)
• Point of external division is given as: \(\left( {x,y} \right) = \left( {\frac{{m{x_2} - n{x_1}}}{{m - n}},\frac{{m{y_2} - n{y_1}}}{{m - n}}} \right)\)

Note: If P is the mid-point of line segment AB, then \(P\left( {x,\;y} \right) = \left( {\frac{{{x_1} + {x_2}}}{2},\frac{{{y_1} + {y_2}}}{2}} \right)\)

CALCULATION:

Here, we have to find the ratio in which the line x + y = 4 divides the line joining the points A (- 1, 1) and B (5, 7).

Let the line x + y = 4 divides the line joining the points A (- 1, 1) and B (5, 7) in the ratio m : 1

Let the point of division be C.

As we know that, the point  internal division is given by: \(\left( {x,\;y} \right) = \left( {\frac{{m{x_2} + n{x_1}}}{{m + n}},\frac{{m{y_2} + n{y_1}}}{{m + n}}} \right)\)

\(⇒ C = \left( {\frac{{5m - 1}}{{m + 1}},\frac{{7m + 1}}{{m + 1}}} \right)\)

∵ C is the point of division i.e C lies on the line x + y = 4 and the coordinates of the point C will satisfy the equation of the line x + y = 4.

\(⇒ \frac{5m - 1}{m + 1} + \frac{7m + 1}{m + 1} =4\)

⇒ (5m - 1) + (7m + 1) = 4(m + 1)

⇒ m = 1/2

So, the required ratio is: (1/2) : 1 = 1 : 2

Hence, option B is the correct answer.

Concept:

Section Formula: Section formula is used to determine the coordinate of a point that divides a line into two parts such that ratio of their length is m : n

1. Let P and Q be the given two points (x1, y1, z1) and (x2, y2, z2) respectively and M(x, y, z) be the point dividing the line segment PQ internally in the ratio m: n

2. Internal Section Formula: When the line segment is divided internally in the ration m: n, we use this formula.\(\rm (x, y, z)=(\frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n}, \frac{mz_2+nz_1}{m+n})\)

Calculation:

Let AB be divided by the plane at R in the ratio k:1, 

Then, coordinates of R are (\(\rm (\frac{3k+1}{k+1},\frac{k+2}{k+1},\frac{2k+3}{k+1})\))

Now, R lies on the plane , so this point must satisfy the equation 2x - y + z = 4

∴ \(\rm \frac{6k+2}{k+1}-\frac{k+2}{k+1}+\frac{2k+3}{k+1}=4\)

\(\rm ⇒ \frac{7k+3}{k+1}=4\)

⇒ 7k + 3 = 4k + 4

⇒ 3k = 1

⇒ k = 1/3

So, the ratio is \(\frac1 3 : 1\)= 1:3

Hence, option (3) is correct. 

Concept:

If P divides the line joining the points A(x₁, y₁, z₁) and B(x₂, y₂, z₂) in the ratio k:1, then the coordinates of P are given by:

P = \(\rm \left(\frac{x_1+kx_2}{k+1},\frac{y_1+ky_2}{k+1},\frac{z_1+kz_2}{k+1} \right )\)

Calculation:

P is a point on the line segment joining points A(3, 2, –1) and B(6, 2, –2).

Let (x1, y1, z1) = (3, 2, –1) and (x2, y2, z2) = (6, 2, –2)

∴ P = \(\rm \left(\frac{3+6k}{k+1},\frac{2+2k}{k+1},\frac{-1-2k}{k+1} \right )\)

According to the question, \(\rm \frac{3+6k}{k+1}\) = 5

⇒ 3 + 6k = 5k + 5

⇒ k = 2

∴ y-coordinate of P 

= \(\rm \frac{2+2k}{k+1}\)

= \(\rm \frac{2+2\times 2}{2+1}\)

= \(\rm \frac{2+4}{3}\)

= 2

∴ The y coordinate of the point P is 2.

The correct answer is option 1.

Concept:

Let P and Q be the given two points (x1, y1) and (x2, y2) respectively, and M be the point dividing the line segment PQ internally in the ratio m : n, then from the section formula, the coordinate of the point M is given by:

\(M(x, y) = \left \{ \left ( \frac{mx_2+nx_1}{m+n} \right ), \left ( \frac{my_2+ny_1}{m+n} \right ) \right \}\)

Let point P be the point that lies at the y-axis and divide the line segment made by two points A and B in the ratio k : 1.

Since point P lies on the y-axis, therefore, the coordinates of the point P would be of the form (0, y).

Now, using the section formula and equating the x-coordinates, we get

\(0 = \frac{3k - 2}{k+1}\)

⇒ 3k - 2 = 0

⇒ k = 2/3

∴ k : 1 = 2 : 3

Hence, the required ratio is 2 : 3.

Concept:

The point of internal division is given as:

 \(\left( {x,\;y} \right) = \left( {\frac{{m{x_2} + n{x_1}}}{{m + n}},\frac{{m{y_2} + n{y_1}}}{{m + n}}} \right)\)

Calculation:

Let the y-axis divides the line joining the points  A(- 6, 15) and B(3, 5) in the ratio m : 1.

Let C be the point of intersection.

As we know that, the point  internal division is given by: 

\(\left( {x,\;y} \right) = \left( {\frac{{m{x_2} + n{x_1}}}{{m + n}},\frac{{m{y_2} + n{y_1}}}{{m + n}}} \right)\)

\(⇒ C = \left( {\frac{{3m - 6}}{{m + 1}},\frac{{5m + 15}}{{m + 1}}} \right)\)

C is the point of division i.e C lies on the y-axis and the equation of the y-axis is x = 0.

So, the point C will satisfy the equation x = 0

⇒ 3m - 6 = 0

⇒ m = 2

So, the required ratio is = 2 : 1

Additional Information

The point of external division is given as:

 \(\left( {x,y} \right) = \left( {\frac{{m{x_2} - n{x_1}}}{{m - n}},\frac{{m{y_2} - n{y_1}}}{{m - n}}} \right)\)

Note: If P is the mid-point of line segment AB, then \(P\left( {x,\;y} \right) = \left( {\frac{{{x_1} + {x_2}}}{2},\frac{{{y_1} + {y_2}}}{2}} \right)\)

Concept:

Section Formula: Section formula is used to determine the coordinate of a point that divides a line into two parts such that ratio of their length is m : n

1. Let P and Q be the given two points (x₁, y₁, z₁) and (x₂, y₂, z₂) respectively and M(x, y, z) be the point dividing the line segment PQ internally in the ratio m: n

2. Internal Section Formula: When the line segment is divided internally in the ration m: n, we use this formula.\(\rm (x, y, z)=(\frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n}, \frac{mz_2+nz_1}{m+n})\)

Calculation:

Here, the point R(5, 4, -6)  divides the points P(3, 2, -4) and Q(9, 8, -10)

Let, required ratio be k:1, Then the coordinates of R are 

(\(\rm \frac{9k+3}{k+1}, \frac{8k+2}{k+1},\frac{-10k-4}{k+1}\))

But coordinates of R are (5, 4, -6)

∴\(\rm \frac{9k+3}{k+1}=5\)

\(\rm ⇒ 9k+3=5k+5\)

 \(\rm ⇒ 4k=2\)

⇒ k = 1/2

∴ k :1 = \(\frac 1 2 :1\)

= 1: 2

Hence, option (2) is correct.

Concept:

Section Formula: Section formula is used to determine the coordinate of a point that divides a line into two parts such that the ratio of their length is m : n

Let A and B be the given two points (x1, y1, z1) and (x2, y2, z2) respectively and C(x, y, z) be the point dividing the line- segment AB internally in the ratio m: n

I. Internal Section Formula: When the line segment is divided internally in the ratio m : n, we use this formula.

 \((x,\ y,\ z) = \ (\frac{mx_2\ +\ nx_1}{m\ +\ n},\ \frac{my_2\ +\ ny_1}{m\ +\ n},\ \frac{mz_2\ +\ nz_1}{m\ +\ n})\)

II. External Section Formula: When point C lies on the external part of the line segment.

 \((x,\ y,\ z) = \ (\frac{mx_2\ -\ nx_1}{m\ -\ n},\ \frac{my_2\ -\ ny_1}{m\ -\ n},\ \frac{mz_2\ -\ nz_1}{m\ -\ n})\)

Calculation:

Let, R divides PQ in the ratio m : n

By using the above formula, the co-ordinate of R is\((\frac{9m+3n}{m+n}, \frac{8m+2n}{m+n}, \frac{-10m-4n}{m+n})\)

But the co-ordinate of R is given (5, 4, -6)

Equating x co-ordinates

\(5=\frac{9m+3n}{m+n}\)

⇒ 5m + 5n = 9m + 3n 

⇒ 4m = 2n

⇒ m/n = 2/4 = 1/2

∴ R divides PQ in the ratio 1 : 2

Given -

The coordinates of the points are:

Point A: (-3, -4) and Point B: (1, 2)

Concept -

The formula to find the coordinates where a line segment is divided by a point (x, y) in the ratio m:n is:

\(\left(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\right) \)

Explanation -

Here, the y-axis intersects the line segment AB at some point (0, y). Let's denote the ratio in which the y-axis divides AB as m:n.

For the y-axis to intersect at (0, y), the x-coordinate will be 0.
Using the section formula:

\( \left(\frac{n \cdot (-3) + m \cdot 1}{m + n}, \frac{n \cdot (-4) + m \cdot 2}{m + n}\right) = (0, y)\)

From this, we get the following equations:

\( \frac{-3n + m}{m + n} = 0 \\ \frac{-4n + 2m}{m + n} = y \)

From the first equation, we get m = 3n.

Substitute m = 3n into the second equation:

\( \frac{-4n + 2(3n)}{3n + n} = y \\ \frac{-4n + 6n}{4n} = y \\ \frac{2n}{4n} = y \\ y = \frac{1}{2} \)

Therefore, the y-axis divides the line segment joining (-3, -4) and (1, 2) in the ratio 3:1.

Concept:

Section Formula: Section formula is used to determine the coordinate of a point that divides a line into two parts such that ratio of their length is m : n

1. Let P and Q be the given two points (x1, y1, z1) and (x2, y2, z2) respectively and M(x, y, z) be the point dividing the line segment PQ internally in the ratio m: n

2. Internal Section Formula: When the line segment is divided internally in the ration m: n, we use this formula.\(\rm (x, y, z)=(\frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n}, \frac{mz_2+nz_1}{m+n})\)

Calculation:

Let, PQ be divided by the xz-plane at a point R in the ratio k:1

Then coordinates of R are (\(\rm (\frac{4k-1}{k+1}, \frac{2k-3}{k+1}, \frac{-k+4}{k+1})\))

Now, R lies on the xz-plane, so y-coordinate will be 0

∴\(\rm \frac{2k-3}{k+1}=0\)

⇒ 2k = 3

⇒ k = 3/2

So, point of interaction = R =  (\(\rm (\frac{4(\frac 3 2)-1}{\frac 3 2+1}, \frac{2(\frac 3 2)-3}{\frac 3 2+1}, \frac{-\frac 3 2+4}{\frac 3 2+1})\))

= (\(\frac{10}{5}, 0, \frac 5 5\))

= (2, 0, 1)

Hence, option (1) is correct.

CONCEPT:

Let A (x₁, y₁) and B (x₂, y₂) be the two given points and the point P (x, y) divide the line joining the points A and B in the ratio m : n, then

• Point of internal division is given as: \(\left( {x,\;y} \right) = \left( {\frac{{m{x_2} + n{x_1}}}{{m + n}},\frac{{m{y_2} + n{y_1}}}{{m + n}}} \right)\)
• Point of external division is given as: \(\left( {x,y} \right) = \left( {\frac{{m{x_2} - n{x_1}}}{{m - n}},\frac{{m{y_2} - n{y_1}}}{{m - n}}} \right)\)

Note: If P is the mid-point of line segment AB, then \(P\left( {x,\;y} \right) = \left( {\frac{{{x_1} + {x_2}}}{2},\frac{{{y_1} + {y_2}}}{2}} \right)\)

CALCULATION:

Let C be the point which divides the line segment joining the points A(5, - 2) and B(9, 6) in the ratio 3 : 1.

Here, x₁ = 5, y₁ = - 2, x₂ = 9, y₂ = 6, m = 3 and n = 1

As we know that, the point of internal division is given by: \(\left( {x,\;y} \right) = \left( {\frac{{m{x_2} + n{x_1}}}{{m + n}},\frac{{m{y_2} + n{y_1}}}{{m + n}}} \right)\)

\(⇒ \left( {x,\;y} \right) = \left( {\frac{{3\cdot {9} + 1 \cdot {5}}}{{3 + 1}},\frac{{3 \cdot {6} - 1 \cdot {2}}}{{3 + 1}}} \right)\)

⇒ (x, y) = (8, 4)

Hence, option A is the correct answer.