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CBSE Sample Paper 2022-23 · Section A · 1 mark

Q18: z-coordinate of the point with x=6x = 6 on line AB

Sample 2022-23Ch 11 · Three Dimensional Geometry
The question

P is a point on the line joining the points A(0,5,−2)A(0, 5, -2) and B(3,−1,2)B(3, -1, 2)⁠. If the x-coordinate of P is 6, then its z-coordinate is

(a) 1010

(b) 66

(c) −6-6

(d) −10-10

Exam answer

The line through A and B is

x−03−0=y−5−1−5=z+22+2\frac{x - 0}{3 - 0} = \frac{y - 5}{-1 - 5} = \frac{z + 2}{2 + 2}
Why?

This is the two-point form of a line, with A as (x1,y1,z1)(x_1, y_1, z_1) and B as (x2,y2,z2)(x_2, y_2, z_2)⁠:

x−x1x2−x1=y−y1y2−y1=z−z1z2−z1\begin{aligned} \frac{x - x_1}{x_2 - x_1} &= \frac{y - y_1}{y_2 - y_1} \\ &= \frac{z - z_1}{z_2 - z_1} \end{aligned}

Watch the signs: z1=−2z_1 = -2⁠, so z−z1=z+2z - z_1 = z + 2 and z2−z1=2+2z_2 - z_1 = 2 + 2⁠.

The three bottoms work out as

3−0=3−1−5=−62+2=4\begin{aligned} 3 - 0 &= 3 \\ -1 - 5 &= -6 \\ 2 + 2 &= 4 \end{aligned}
or x3=y−5−6=z+24=k\text{or } \frac{x}{3} = \frac{y - 5}{-6} = \frac{z + 2}{4} = k

Any point on it is (3k,5−6k,4k−2)(3k, 5 - 6k, 4k - 2)⁠.

Why?

Call the common value of the three fractions kk⁠. Then x=3kx = 3k⁠, y−5=−6ky - 5 = -6k and z+2=4kz + 2 = 4k⁠. Solving each for the coordinate gives the general point:

x3=k⇒x=3ky−5−6=k⇒y−5=−6k⇒y=5−6kz+24=k⇒z+2=4k⇒z=4k−2\begin{aligned} \frac{x}{3} = k &\Rightarrow x = 3k \\ \frac{y - 5}{-6} = k &\Rightarrow y - 5 = -6k \\ &\Rightarrow y = 5 - 6k \\ \frac{z + 2}{4} = k &\Rightarrow z + 2 = 4k \\ &\Rightarrow z = 4k - 2 \end{aligned}

Each value of kk gives one point of the line: k=0k = 0 gives A and k=1k = 1 gives B.

3k=6⇒k=23k = 6 \Rightarrow k = 2⁠, so the z-coordinate of P is 4(2)−2=64(2) - 2 = 6⁠.

Why?

P is on the line, so it is (3k,5−6k,4k−2)(3k, 5 - 6k, 4k - 2) for some kk⁠. Its x-coordinate is 6, which fixes k=2k = 2⁠. Then put k=2k = 2 into the z-coordinate.

3k=6k=63=2z=4(2)−2=8−2=6\begin{aligned} 3k &= 6 \\ k &= \frac{6}{3} = 2 \\[4pt] z &= 4(2) - 2 \\ &= 8 - 2 = 6 \end{aligned}

The whole point is P(6,−7,6)P(6, -7, 6)⁠.

y=5−6(2)=5−12=−7\begin{aligned} y &= 5 - 6(2) \\ &= 5 - 12 = -7 \end{aligned}

Answer: (b) 66

Where this comes from

Chapter 11, Three Dimensional Geometry: Point on the line through two points.

Question type: The line through two points

How to spot it: click to see

Type 2. The line through two points: subtract, then divide

Spot it: two points are given, and you need the direction of the line through them.

Here: B minus A gives 3,−6,43, -6, 4⁠, so the line is x3=y−5−6=z+24=k\frac{x}{3} = \frac{y - 5}{-6} = \frac{z + 2}{4} = k⁠. Its points are (3k,5−6k,4k−2)(3k, 5 - 6k, 4k - 2)⁠, and x=6x = 6 gives k=2k = 2⁠.

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