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Question

In what direction a line be drawn through the point (1, 2) so that its points of intersection with the line x+y=4 is at a distance 63 from the given point

A
30
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B
45
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C
60
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D
75
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Solution

The correct option is D 75
Let the required line through the point (1,2) be inclined at an angle θ to the axis of x. Then its equation is
x1cos θ=y2sin θ=r .....(i)
where r is distance of any point (x, y) on the line from the point (1, 2).
The coordinates of any point on the line (i) are (1+r cos θ,2+r sin θ). If this point is at a distance 63 form (1, 2), then r=63.
Therefore, the point is (1+63cos θ,2+63sin θ) .
But this point lies on the line x+y=4.
63(cos θ+sin θ)=1orsin θ+cos θ=36
12sin θ+12cos θ=32,
{Dividing both sides by 2}
sin(θ+45)=sin60 or sin120
θ=15 or 75.

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