Straight lines practice · Medium

Option A (9) is correct because the centroid's coordinates are (4, 5), and (h+k)/(k-h) = (4+5)/(5-4) = 9.

The correct answer is A (x > 8/15, y < -8/5) because the point P(x, y) lies on the line y = -3x and is on the opposite side of the line 3x - 4y - 8 = 0 as the point (3, 4).

Option C ($2 \sqrt{2}$) is correct because the least value of $|a|$ is found by ensuring the discriminant of the quadratic equation in 'm' is non-negative.

Option A ($x+4=0$) is correct because the tangents intersect at right angles, which implies that the directrices of the parabolas intersect at the point of intersection of the tangents. The directrices are x = -4 and the intersection point lies on the line x = -4 or x + 4 = 0.

Option C (obtuse angled) is correct because the cosine of one of the angles is negative, indicating an angle greater than 90 degrees.

Option A (a straight line) is correct because the locus simplifies to a linear equation.

The correct answer is A (21x + 77y - 101 = 0) because it represents the bisector of the obtuse angle between the two lines, determined by checking the sign of a1a2 + b1b2.

Option B ($2 T x-a^2 y+2 a T=0$) is correct because it represents the equation of the line passing through (-a, 0) and forming a triangle of area T with the axes.

Option B ($(3,1),(-7,11)$) is correct because these points lie on the line $x+y=4$ and are at a unit distance from the line $4x+3y=10$.

Option B (Circle) is correct because the locus of point P, which divides the line segment in a given ratio, forms a circle.

Option A ($\left(-\frac{\pi}{4}, \frac{\pi}{4}\right) \cup\left(\frac{\pi}{4}, \frac{3 \pi}{4}\right)$) is correct because it represents the interval where both points A and B lie on the same side of the line x-y=0.

Option A ($a+b-c>0$) is correct because it is derived from the distance formula and the given condition that the distance between (1,1) and the intersection point is less than $2\sqrt{2}$.

Option C ($x-2=0,3 x+4 y=18$) is correct because these lines satisfy the given conditions of passing through (2,3) and intercepting a length of 2 units between the given lines.

Option A (${x^2} + 3{y^2} - 4xy - 14x + 24y + 45 = 0$) is correct because it represents the joint equation of the pair of lines passing through (3, -2) and parallel to the given equation.

The correct answer is A (3x - y = 7 and x + 3y = 9) because the equations of lines passing through (3, 2) and making an angle of 45 degrees with the line x - 2y = 3 are found using the angle between two lines formula.

Option C ($\frac{{x}}{{10}}+\frac{{y}}{{2}}=-1$ and $\frac{{x}}{{4}}+\frac{{y}}{{12}}=1$) is correct because these are the equations of the lines that satisfy the given conditions.

Option D ($5x-2y-7=0$) is correct because it represents the line passing through the intersection points of both families of lines.

Option B ($\frac{x}{a} - \frac{y}{b} = 1$) is correct because it represents the equation of the line $l$ that is perpendicular to the given line and passes through the x-intercept of the given line.

Option C ($\frac{{33}}{{10}}$) is correct because it is the distance between the parallel lines calculated using the appropriate formula.

The correct answer is A (${x^2} \pm Ax \mp 2A = 0$) because it represents the quadratic equation whose roots are the x and y intercepts of the line, derived from the given area and point (2,2).

Option D is correct. The combined equation of the lines through (-1,-1) and making an angle of 45 degrees with x+y=0 is found to be $xy + x + y + 1 = 0$.

Option A (4) is correct because there are four lines equally inclined to both axes, one in each quadrant.

Option D ($5x - 2y - 7 = 0$) is correct because it is the equation of the line passing through the intersection points of the two families of lines.

Option A ((-1,1),(1,-1)) is correct because the given condition leads to two linear equations, and comparing them with the family of lines yields the points of concurrency.

Option D ($x^2+y^2=13$) is correct because the variable line always passes through a fixed point (2,-3), which lies on the circle $x^2+y^2=13$.

Option A (5) is correct because solving for the angular bisector of the internal angle B leads to the equation 3x + y - 5 = 0, thus k = 5.

Option B (18) is correct because the minimum value of OA + OB is found by minimizing the function representing the sum of the intercepts.

Option C ($a = 1 + b$) is correct because it is one of the conditions derived from setting the determinant of the matrix formed by the points to zero.

The correct answer is D ($\frac{a+b}{1-a b}$) because it is derived from the trigonometric identity for the tangent of a sum of angles.

Option B ($\frac{1}{{{x^2}}} + \frac{1}{{{y^2}}} = \frac{4}{{{p^2}}}$) is correct because it represents the locus of the midpoint derived from the given line equation.