Question

Write an equation of a line that passes through the point (2,0) and is perpendicular to the line y=−$\frac{1}{2}$x+3.
y= [ ]

Answer 1

Answer:

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Step-by-step explanation:

ANSWER

Since, line of shortest distance is perpendicular to both the lines, its direction ratios can be obtained by cross-product of direction ratios of the two lines.

(4

i

^

−2

j

^

)×(5

i

^

+3

j

^

)=22

k

^

Direction of line of shortest distance=

k

^

Let

4

x+4

=

−2

y−2

=

0

z−3

=a

and

5

x−5

=

3

y−3

=

0

z

=b

Point of contact of first line and line of shortest distance =(4a−4,−2a+2,3)

Point of contact of second line and line of shortest distance =(5b+5,3b+3,0)

Since, line of shortest distance is perpendicular to both the lines,

4(4a−5b−9)−2(−2a−3b−1)=0⇒10a−7b=17

5(4a−5b−9)+3(−2a−3b−1)=0⇒7a−17b=24

On solving, we get a=1,b=−1

Substituting a=1, we get a point (0,0,3) that lies on the line of shortest distance

So, equation of line of shortest distance :

0

x

=

0

y

=

1

z−3