Straight lines practice · Mixed difficulty

Option A ($-\frac{{4 \sqrt{{2}}}}{{3}} < t < \frac{{5 \sqrt{{2}}}}{{6}}$) is correct because it defines the range of values for $t$ where the point P lies between the two given parallel lines.

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 A ($- \pi /4,\pi /4$) is correct because it represents the interval where both points P and Q lie on the same side of the line x - y = 0, satisfying the given conditions.

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.

The correct answer is C (0,0) because the perpendicular distance from (0,0) to each of the given lines is the same.

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.