Straight lines practice · Easy

Option A ($\left( { - 8, - 14} \right)$) is correct because it represents the harmonic conjugate of the given point with respect to the other two points, calculated using the section formula.

Option C (There will be a set of parallel lines) is correct because the equation represents a family of lines with the same slope 'm' but different y-intercepts.

Option C (Both are true) is correct because both the distance formula and the midpoint formula are stated accurately.

Option C ($(3 x-1)^2+(3 y)^2=a^2+b^2$) is correct because it represents the locus of the centroid of the triangle with the given vertices.

Option C ($y = \sqrt 3 x + \lambda$) is correct because it represents a line making a 30-degree angle with the given line.

Option D (None of these) is correct because the calculated area of the triangle with vertices (-8, 0), (9, 11), and (-1, -11) is 66, which is not among the given options.

Option A ((3, -1)) is correct because the family of lines passes through the intersection of the two lines formed by separating the lambda and mu terms.

Option D (Below the x-axis at a distance of (3/2) from it) is correct because the equation of the line parallel to the x-axis is y = -3/2.

The correct answer is A (2401) because it is the value of $t^4$ after solving the system of equations and finding t.

The correct answer is B (-1) because this value of m makes the line y=mx pass through the intersection point of the other two lines.

Option D (HP for all $m$) is correct because the condition for concurrency leads to a relationship between a, b, and c that satisfies the definition of a Harmonic Progression.

Option D ($L_{1} \perp L_{2}, L_{1} \perp L_{3}, L_{2}$ intersect $L_{4}$) is correct because it accurately describes the relationships between the given lines based on their slopes and intersection points.

The correct answer is B (-4) because it satisfies the condition that the perpendicular bisector of the line segment joining P and Q has a y-intercept of -4.

Option C (HP) is correct because $AC, AB, AD$ satisfy the condition for Harmonic Progression, which is $\frac{1}{{AC}} + \frac{1}{{AD}} = \frac{2}{{AB}}$.

Option C ($\frac{{35}}{{3\sqrt {34} }}$) is correct because it is the result of applying the formula for the distance between parallel lines after normalizing the equations.

Option C ($x + 3y + 14 = 0$) is correct because it represents the equation of a line that passes through the point (-2, -4) and is perpendicular to the line $3x - y + 5 = 0$.

Option A ($\left( {0,\\,\\,\frac{\pi }{2}} \right)\,\,$) is correct because it represents the interval where both points lie on the same side of the line x + y = 1, derived from the inequality sin(theta) + cos(theta) > 1.

Option C (A square) is correct because the locus forms a square with vertices at (±a, ±a).

Option D (None of these) is correct because the side lengths of the triangle formed by the given lines are not equal, nor do they satisfy the Pythagorean theorem, indicating it is neither isosceles, equilateral, nor right-angled.

Option B (-1/3) is correct because it satisfies the condition for the three points to be collinear.

Option C is correct because both statements are fundamental concepts in coordinate geometry.

Option C ($x + y + 1 = 0$) is correct because it passes through the point (1, -2) and has equal intercepts.

The correct answer is B (lie on a straight line) because if x and y coordinates are in GP with the same common ratio, the slope between any two points will be the same, implying collinearity.

Option D (if $a, \frac{{2 c}}{5}, b$ are in HP) is correct because the collinearity condition leads to a harmonic progression relationship between a, b, and c.

Option B (9/2 sq unit) is correct because it represents the area of the triangle formed by the given lines, calculated using the coordinates of the intersection points.

Option C (20) is correct because it is found by determining the equation of the angle bisectors and comparing coefficients.

Option D ($\left( {\frac{{13}}{4},\frac{9}{4}} \right)$) is correct because by finding the equations of the perpendicular bisectors of two sides and solving them simultaneously, we can find the circumcenter.

Option B (5/2) is correct because for the points to be collinear, the slope between any two pairs of points must be equal, leading to a=5/2.

Option D ($4x + 3y = 0$) is correct because it is the equation of the line passing through the origin and the intersection point of the two given lines.

Option D (a<7 or a>11) is correct because for the points to be on the same side of the line, the product of the expressions obtained by substituting the points into the line equation must be greater than 0, leading to the inequality (a-7)(a-11) > 0.