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Straight Lines

Multiple Choice Questions

221.

The orthocentre of the triangle formed by the lines x + y = 1 and 2y² - xy - 6x² = 0

$(\frac{4}{3}, \frac{4}{3})$
$(\frac{2}{3}, \frac{2}{3})$
$(\frac{2}{3}, \frac{- 2}{3})$
$(\frac{4}{3}, \frac{- 4}{3})$

222.

The incentre of the triangle formed by the straight lines $y = \sqrt{3} x, y = - \sqrt{3} x and y = 3 is$ and y = 3 is

(0, 2)
(1, 2)
(2, 0)
(2, 1)

223.
The points on the straight line 3x - 4y + 1 = 0 which are at a distance of 5 units from the point (3, 2) are

$(- 2, - \frac{7}{4}) (- 3, \frac{- 5}{2})$

$(4, \frac{11}{4}) (- 1, - 1)$

$(1, \frac{1}{2}) (2, \frac{5}{4})$

$(7, 5), (- 1, - 1)$

$(7, 5), (- 1, - 1)$

$(d) We have, line 3 x - 4 y - 1 = 0 and point A (x_{1}, y_{1}) which is at 5 units distance from (3, 2) equations of line PA are$

$\frac{x_{1} - 3}{\cos (θ)} = \frac{y_{1} - 2}{\sin (θ)} = \pm 5 \Rightarrow x_{1} = 3 \pm 5 \cos (θ) y_{1} = 2 \pm 5 \sin (θ) Since, (x_{1}, y_{1}) lies on the 3 x - 4 y - 1 = 0 ∴ 3 (3 \pm 5 \cos (θ)) - 4 (2 \pm 5 \sin (θ)) - 1 = 0 \Rightarrow 9 \pm 15 \cos (θ) - 8 + 20 \sin (θ) - 1 = 0 \Rightarrow 15 \cos (θ) - 20 \sin (θ) = 0 \Rightarrow 3 \cos (θ) = \pm 4 \sin (θ) \Rightarrow \tan (θ) = \pm \frac{3}{4} \cos (θ) = \pm \frac{4}{5} \sin (θ) = \pm \frac{3}{5} ∴ x_{1} = 3 \pm (\frac{4}{5}) 5 = 7, - 1 y_{1} = 2 \pm 5 (\frac{3}{5}) = 5, - 1 ∴ Coordinates are (7, 5) and (- 1, - 1)$

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Multiple Choice Questions

223. The points on the straight line 3x - 4y + 1 = 0 which are at a distance of 5 units from the point (3, 2) are - 2, - 74 - 3, - 52 4, 114 - 1, - 1 1, 122, 54 7, 5, - 1, - 1

223.
The points on the straight line 3x - 4y + 1 = 0 which are at a distance of 5 units from the point (3, 2) are

$(- 2, - \frac{7}{4}) (- 3, \frac{- 5}{2})$

$(4, \frac{11}{4}) (- 1, - 1)$

$(1, \frac{1}{2}) (2, \frac{5}{4})$

$(7, 5), (- 1, - 1)$