First slide

Equation of line in Straight lines

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

Moderate

A

PR = 3 sec θ
B

PS = 4 cosec θ
C

$P R + P S = \frac{2 (3 \sin θ + 4 \cos θ)}{\sin 2 θ}$
D

$\frac{9}{{(P R)}^{2}} + \frac{16}{{(P S)}^{2}} = 1$

Solution

The equation of the line in parametric form is
$\frac{x - 3}{\cos θ} = \frac{y - 4}{\sin θ} = r$
Any point on this line is (3 + r cos θ, 4 + r sin θ)
It lies on x = 6 if 3 + r cos θ = 6 ⇒ r = 3 sec θ
∴ PR = 3 sec θ
Again, the point lies on y = 8 if 4 + r sin θ = 8
∴ r = 4 cosec θ or PS = 4 cosec θ

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