Straight lines practice · Easy

Option A (1) is correct because setting the slope formula equal to 2 and solving for x yields x = 1.

Option D (63) is correct because it is the area of the pentagon calculated using the coordinate geometry formula.

Option C ($\sqrt{40}$) is correct because it is the distance calculated between the point (1,2) and the intersection point (-1,-4) along the specified parallel line.

Option A ($\pi / 6$) is correct because the acute angle between the lines is calculated to be 30 degrees, which is equivalent to $\pi/6$ radians.

Option D ($\frac{\pi}{4}$) is correct because the angle between the lines is calculated using the slopes of the lines.

Option A (0) is correct because the maximum value of $\sin \theta$ is 0 when $\theta$ lies in the interval $[\frac{{4\pi}}{{3}}, 2\pi]$ based on the given inequality.

Option B ($a, c, b$ are in G.P.) is correct because the area of the triangle formed by the line and the axes is constant only when $a, c, b$ are in geometric progression.

Option B ($(2,4 / 3)$) is correct because it is the result of applying the section formula to find the coordinates of point P.

Option C (The orthocenter) is correct because the lines x - 3y = 0 and 3x + y = 0 are perpendicular, making the origin the orthocenter, and the line 3x - 4y = 0 passes through it.

Option C (6) is correct because the determinant of the coefficients of the concurrent lines must be zero, leading to k = 6.

Option A ($\frac{\pi}{2}$) is correct because the lines are the x-axis and y-axis, which are perpendicular.

Option B ($x \cos 120^{\circ}+y \sin 120^{\circ}=4$) is correct because it represents the given equation in normal form, where the coefficients of x and y are cosine and sine of the angle, respectively, and the right-hand side is the perpendicular distance from the origin.

Option D, $y = \sqrt{{3}}x$, is correct because it represents a line passing through the origin and making a 30-degree angle with the given line.

Option A ($2x - 3y + 1 = 0$) is correct because it represents a line passing through the intersection of the given lines and is perpendicular to the line $3x + 2y + 5 = 0$.

Option C $\left( { - \frac{{1}}{{\sqrt 2 }},\frac{{7}}{{\sqrt 2 }}} \right)$ is correct because rotating the point P(4, 3) by 45 degrees counter-clockwise about the origin results in the new coordinates, which can be found using rotation transformation formulas.

Option A ($2x + 5y + 17 = 0$) is correct because it represents a line perpendicular to the given line and passes through the intersection point of the other two lines.

Option D ($x + y - 7 = 0$) is correct because the diagonal BD is perpendicular to AC and passes through the midpoint of AC.

Option B ($\frac{1}{4}$ sq unit) is correct because it is the area of the triangle formed by the intersection of the given lines.