Pair of straight lines practice · Mixed difficulty

Option C ($\tan^{{-1}}(\frac{{7}}{{4}})$) is correct because it is the angle calculated using the formula for the angle between two lines represented by a second-degree equation.

Option D ($p \leq \alpha$ or $p \geq \beta$) is correct because the condition for the equation to represent a pair of straight lines leads to the inequality $(p-\alpha)(p-\beta) \geq 0$, which implies $p \leq \alpha$ or $p \geq \beta$.

Option A (4) is correct because substituting the coefficients into the condition for a pair of straight lines and solving for $k$ yields $k=4$.

Option D (all values of $\lambda$) is correct because the homogeneous equation of the lines joining the origin to the intersection points results in a pair of perpendicular lines, independent of $\lambda$.

Option A ($(4-\sqrt{5}) x-(3-2 \sqrt{5}) y+(2-4 \sqrt{5})=0$) is correct because it represents the equation of the bisector of the obtuse angle between the two given lines, derived using the angle bisector formula and considering the condition for obtuse angle.

Option D (3) is correct because it satisfies the condition for the equation to represent a pair of lines.

Option B ($\sqrt {2/3}$) is correct because it represents the length of each side of the equilateral triangle given the vertex and the base.

Option C ($b x^2-2 h x y+a y^2=0$) is correct because it represents the equation of the pair of straight lines through the origin and perpendicular to the given lines.

Option D (None of these) is correct because the equation $y^3-4 x^2 y=0$ represents three concurrent lines passing through the origin, and thus does not form a triangle.

Option B ((1,-2)) is correct because it is the point of intersection found by solving the partial derivative equations of the given equation with respect to x and y.

Option C ($\sqrt {\frac{{5}}{{2}}}$) is correct because the distance between the parallel lines is calculated using the formula $2\sqrt {\frac{{{{g^2} - ac}}}{{a\left( {a + b} \right)}}}$.

Option C ($a=\frac{64}{115}, b=-\frac{8}{115}$) is correct because the line $ax + by = 1$ must be one of the angle bisectors of the given lines.

Option C, $h(1-m^2)+m(a-b)=0$, is correct because it represents the condition when $y=mx$ is one of the angle bisectors of the given pair of lines.

The correct answer is D (None of these) because the equation of the lines parallel to the given lines and passing through (x1, y1) is $a(x-x_1)^2+2h(x-x_1)(y-y_1)+b(y-y_1)^2=0$. The provided answer in the solution steps is also incorrect.

Option B ($2{x^2} - 3xy + 2{y^2} = 0$) is correct because it represents the equation of the pair of straight lines perpendicular to the given pair and passing through the origin.

Option C ($21x + 77y - 101 = 0$) is correct because it is the equation of the bisector of the acute angle between the two lines, determined by checking the sign of $a_1a_2 + b_1b_2$.

Option A ($6x + y - 32=0$) is correct because it represents the locus of the point such that the area of the triangle formed by it with the points (1, 5) and (3,-7) is 21 sq unit.

Option D ($\frac{{2h}}{{a-b}}$) is correct because it represents the formula for the angle of rotation required to eliminate the xy term in a general conic equation.

Option D (A point) is correct because the equation can be rewritten as the sum of two squares equal to zero, which only holds true for a single point.

Option C ((0,0)) is correct because the triangle formed by the lines is a right-angled triangle at the origin, and the orthocenter of a right-angled triangle is the vertex at the right angle.

Option B ($5 h^2=9 a b$) is correct because the condition for one slope being 5 times the other is derived using the sum and product of slopes of the lines represented by the given equation.

Option D ($h^2=a^2$) is correct because for coincident lines, the condition is that the discriminant of the quadratic equation must be zero, leading to $h^2 = a^2$.

Option C is correct because after rationalizing and simplifying the limit, the expression converges to 1/(4√2) as y approaches infinity.

Option C (2) is correct because when $\lambda = 2$, the determinant of the matrix formed by the coefficients of the given equation is zero, indicating that the equation represents a pair of straight lines.

Option A (They are equally inclined to one another) is correct because the equation represents three straight lines passing through the origin that are equally inclined to one another, derived from trigonometric identities.

Option A (0) is correct because substituting the coefficients into the equation and solving for $\lambda$ yields $\lambda = 0$.

Option C ($g^2-f^2=2 c$) is correct because the condition for the lines joining the origin to the intersection points to be at right angles leads to the equation $g^2 - f^2 = 2c$.

Option C ($y=x$) is correct because it is the third line when two of the three lines represented by $x^3+a x^2 y+b x y^2+y^3=0$ are perpendicular.

Option A (a straight line) is correct because the locus of R, derived from the given harmonic mean condition, results in a linear equation.

Option A (a pair of straight lines) is correct because the discriminant $abc + 2fgh - af^2 - bg^2 - ch^2 = 0$ for the given equation.