hetmavani wrote:

In the xy-plane, l and m are two lines. (3, 4) is a point only on one among l and m and (4, 5) lies only on the other. Are lines l and m perpendicular?

(1) The slope of line l is −1.

(2) They intersect at (2, 1).

OA is D.

as per question , Points are (3,4) , (4,5). Only one point of these lie on line l, other has be on m.

Statement 1 : The slope of line l is −1.

Case 1if (3,4) falls on line l ,then equation of line l will be

\(\frac{(y-4)}{(x-3)}\)=-1

y=-x+7

Assume that line m passing through (4,5) is perpeniducular to line l

As product of slope of two non vertical perpendicular line is -1.

Slope of line m would be 1

\(\frac{(y-5)}{(x-4)}\)=1

y=x+1

Now the point of interesection of line l and m would come out be (3,4).

As per question (3,4) should not lie on line m.

so Slope of m cannot ever be -1.

This tell us that l cannot be perpendicular to m.

Case 2if (4,5) falls on line l ,then equation of line l will be

\(\frac{(y-5)}{(x-4)}\)=-1

y=−x+9

Assume that line m passing through (3,4) is perpeniducular to line l

As product of slope of two non vertical perpendicular line is -1.

Slope of line m would be 1

\(\frac{(y-4)}{(x-3)}\)=1

y=x+1

Now the point of interesection of line l and m would come out be (4,5).

As per question (4,5) should not lie on line m.

so Slope of m cannot ever be -1.

This tell us that l cannot be perpendicular to m.

So Statement 1 alone is sufficient to give a definite answer(i.e No) for question:

Are lines l and m perpendicular?.

Statement 2 : line l and m interesect at (2,1)

Now we have got two points on each line , so each of these two line can be defined.

We do not need nomenclature l,m.To avoid confusion,We can just name them as line 1 and line 2

Line 1 passing through point (3,4) and (2,1)

\(\frac{(y−4)}{(x−3)}\)=\(\frac{(1-4)}{(2-3)}\)

y=3x−5

Line 2 passing through point (4,5) and (2,1)

\(\frac{(y−5)}{(x−4)}\)=\(\frac{(1-5)}{(2-4)}\)

y=2x−3

For two non vertical line to be perpendicular , product of their slope should be -1.

In case of line 1 and line 2 , product of their slope is 3∗2= 6

Hence , we are getting a definite answer(i.e No) for question:

Are lines l and m perpendicular?Statement 2 alone is sufficient

As Statement 1 and Statement 2 are both alone sufficient , OA should be D

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