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Question

A line which makes an acute angle θ with the positive direction of x-axis is drawn through the point P(3, 4) to meet the line x = 6 at R and y = 8 at S, then



Correct Answer
A
PR=3 sec θ
Correct Answer
B
PS=4 cosecθ
Correct Answer
C
PR+PS=2(3 sin θ+4 cos θ)sin 2θ
Correct Answer
D
9(PR)2+16(PS)2=1

Solution

The correct options are
A PR=3 sec θ
B PS=4 cosecθ
C PR+PS=2(3 sin θ+4 cos θ)sin 2θ
D 9(PR)2+16(PS)2=1

Equation of any line making an acute angle θ with positive direction of x-axis and passing through P(3, 4) is

x3cos θ=y4sin θ=r . . .(i)

where |r| is the distance of any point (x, y) from P.

Therefore, A(r cosθ+3,rsinθ+4) is a general point on line (1).

If A is R, r cos θ+3=6r=3cos θ=3 sec θ

Since θ is acute, cos θ>0

    PR=r=3 sec θ . . . (ii)

If A = S, r sin θ+4=8,

   r=4 cosec θ,

PS = 4 cosec θ,

Also PR+PS=3cos θ+4sin θ

=2(3 sin θ+4 cos θ)sin 2θ and 9(PR)2+16(PS)2=cos2θ+sin2θ=1

(A), (B), (C) and (D) all are correct.

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