Pair of straight lines practice · Mixed difficulty

Option A (2) is correct because it satisfies the given condition for the angle between the pair of straight lines after substituting the values into the relevant formula and solving the resulting quadratic equation.

Option A ($x-3 y=30$) is correct because it represents the equation of a line passing through (3,-9) and perpendicular to the line joining the origin and (3,-9).

The correct answer is None of these because the combined equation is $3x^2 - 8xy - 3y^2 - 4x + 22y - 7 = 0$, which doesn't match any of the options.

Option D (-9) is correct because the condition for a general quadratic equation to represent a pair of lines is applied.

Option A (5/2) is correct because the lines are parallel, and the distance is calculated using the formula for distance between parallel lines.

Option C ($4 x^2-4 x y+y^2=4$) is correct because it represents a pair of straight lines when the determinant ($\Delta$) of the general second-degree equation is zero.

Option A (($x y_1-y x_1)^2=d^2(x^2+y^2)$) is correct because it represents the equation of the pair of lines passing through the origin, given the distance 'd' from the point (x1, y1).

Option D (None of these) is correct because the correct equation of the pair of straight lines drawn through (1,1) and perpendicular to the pair of lines $3 x^2-7 x y+2 y^2=0$ is $2(x-1)^2+7(x-1)(y-1)+3(y-1)^2=0$, which is not listed.

Option A (1/5) is correct because it is the value of m calculated using the formula for the angle between two lines represented by a homogeneous equation.

Option C (3) is correct because the determinant of the coefficients of the concurrent lines must be zero, leading to q = 3.

Option B ($6 x-5 y+14=0$) is correct because it represents the equation of the diagonal AC, which is nearer to the origin compared to diagonal BD.

Option B ($h=0$) is correct because if the coordinate axes are the bisectors, the $xy$ term must vanish, implying $h=0$.

Option D ($\frac{c}{\sqrt{(a-b)^2+4 h^2}}$) is correct because it represents the product of the perpendiculars from the origin to the lines represented by the given equation.

Option D ($a=11, b=-11$) is correct because the equation represents a pair of perpendicular lines if the sum of the coefficients of $x^2$ and $y^2$ is zero (a + b = 0).

Option D (-3) is correct because substituting the slope of the given line into the equation allows us to solve for c.

Option B ($2\sqrt 2$) is correct because the length of the perpendicular from the origin to the normal at (2,2) is the distance between (0,0) and (2,2).

Option B ($\pm \frac{3}{2}$) is correct because the ratio of slopes is used to find a relationship between h and the slopes, which then leads to the value of h.

The correct answer is $x+5y=10$ because it represents a line perpendicular to $y=5x$ that forms a triangle with the axes having an area of 10 square units.

Option B (4) is correct because when k=4, the given equation represents a pair of straight lines as the determinant condition is satisfied.

Option A (0.2) is correct because it is the absolute value of the tangent of the angle between the lines, calculated using the formula involving the coefficients of the quadratic equation.

Option D ($\lambda$ may have any value) is correct because the condition for perpendicularity is always satisfied regardless of the value of $\lambda$.

Option A (square) is correct because all sides are equal in length, and the diagonals are perpendicular, indicating 90-degree angles.

Option B ($3 x-2 y+3=0$) is correct because it represents the equation of the reflected ray, calculated by finding the intersection point and using the angle of incidence equals angle of reflection principle.

Option A (Two of them are coincident and two of them are perpendicular) is correct because the equation $x^3+x^2 y-x y^2-y^3=0$ represents three straight lines, where two are coincident and two are perpendicular.