The coordinates of the point where the line through the points A(3,4,1) and B(5,1,6) crosses the XY-plane are

Correct answer: A. ((13)/(5), (23)/(5), 0) Equation of the line through the given points is (x-3)/(5-3)=(y-4)/(1-4)=(z-1)/(6-1) (x-3)/(2)=(y-4)/(-3)=(z-1)/(5) Any point on this line can be taken as (3+2 λ, 4-3 λ, 1+5 λ) If this point lies on XY-plane then the z-coordinate is zero 1+5 λ=0 λ=- (1)/(5) Thus the required coordinates of the point are ( 3- (2)/(5), 4-3(-(1)/(5)), 0) ((13)/(5), (23)/(5),0)

BITSAT 2013

Exam: BITSAT
Year: 2013
Subject: Mathematics
Chapter: Three Dimensional Geometry

BITSAT 2013 · Mathematics · Three Dimensional Geometry

The coordinates of the point where the line through the points...

The coordinates of the point where the line through the points

\mathrm{A}(3,4,1)

and

\mathrm{B}(5,1,6)

crosses the XY-plane are

Options

A $\left(\frac{13}{5}, \frac{23}{5}, 0\right)$
B $\left(-\frac{13}{5}, \frac{23}{5}, 0\right)$
C $\left(\frac{13}{5},-\frac{23}{5}, 0\right)$
D $\left(-\frac{13}{5}, \frac{-23}{5}, 0\right)$

Correct answer

A. $\left(\frac{13}{5}, \frac{23}{5}, 0\right)$

Step-by-step solution

Equation of the line through the given points is

\frac{x-3}{5-3}=\frac{y-4}{1-4}=\frac{z-1}{6-1}\Rightarrow \frac{x-3}{2}=\frac{y-4}{-3}=\frac{z-1}{5}

Any point on this line can be taken as

(3+2 \lambda, 4-3 \lambda, 1+5 \lambda)

If this point lies on XY-plane then the z-coordinate is zero \Rightarrow 1+5 \lambda=0 \Rightarrow \lambda=-

\frac{1}{5}

Thus the required coordinates of the point are

\left(

\frac{2}{5},

4-3\left(-\frac{1}{5}\right),

0\right)

\equiv\left(\frac{13}{5}, \frac{23}{5},0\right)

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