Straight lines practice · Mixed difficulty

Option D (4/3) is correct because the y-intercept is found by substituting x=0 into the equation of the line perpendicular to the given line and passing through (2,2).

The correct answer is C ($y=5$) because the locus of vertex C is found by expressing the coordinates of C in terms of a parameter and eliminating it.

Option C ($p^3+q^3+r^3=3 p q r$) is correct because the concurrency condition simplifies to this relationship, which is equivalent to $p+q+r=0$ or $p=q=r$. Since p, q, r are distinct, $p+q+r=0$ must hold.

Option B ($\frac{60}{7}$) is correct because it represents the distance PQ calculated by finding the coordinates of P and Q using the given information about the midpoint and the lines they lie on.

Option D ($\sqrt{\frac{2}{5}}$) is the correct answer because after finding the value of 'a' using the perpendicularity condition, we solve for the intersection point and calculate its distance from the origin.

Option D ($\tan ^{-1}(-2)$) is correct because it is derived from the relationship between the sum of slopes and the coefficient of the xy term in the given equation.

Option B (0) is correct because substituting the expression for p in terms of a and b, and using the A.P. condition, simplifies the given expression to zero.

Option B is correct. The figure formed by $a|x|+b|y|+c=0$ is a rhombus, and its area is calculated as $\frac{2c^2}{|ab|}$.

Option D ($x - 3y + 5 = 0$) is correct because it represents a line passing through the intersection point and perpendicular to one of the given lines.

The correct answer is A ((0,π/2)) because it's the range of theta values for which both points lie on the same side of the line, determined by the sign of the expression.

Option D (($1,\frac{1}{{\sqrt 3 }}$) is correct because the triangle is equilateral, and the incenter coincides with the centroid, which is easily calculated.

Option A (1) is correct because when the points are collinear, the slope between any two pairs of points must be equal, allowing us to solve for x.

Option C (H.P.) is correct because the concurrency condition leads to the relationship $b = \frac{{2ac}}{{a+c}}$, which is the definition of Harmonic Progression.

Option D ($x + y + \frac{1}{{\sqrt 2 }} = 0$) is correct because it represents a line parallel to BC at a distance of 1/2 from the origin.

Option C ($x + 7y = 31$) is correct because it represents the equation of the second diagonal which is perpendicular to the given diagonal and passes through the given vertex.

Option A ($2 x+3 y=18$) is correct because it is the equation of the perpendicular bisector of the line segment AB, found by calculating the midpoint and slope of AB and then using the point-slope form.

Option B ($\left(\frac{3}{5}, \frac{{-2}}{5}\right)$) is correct because it satisfies the equation derived from the given condition $3a - 2b + 5c = 0$.

Option B (-7) is correct because the collinearity condition (zero area of the triangle formed by the points) leads to λ = -7.

Option A ($\left( {\frac{{11}}{5},\,\frac{{28}}{5}} \right)$) is correct because it is the result of applying the formula for finding the image of a point with respect to a line.

Option A ($-\frac{{a \alpha+b \beta+c}}{{a \cos \theta+b \sin \theta}}$) is correct because it represents the length PQ derived from the equation of a line and its intersection with another line.

Option D ($x - 1 = 0, y - 1 = 0$) is correct because these are the equations of lines passing through (1,1) at a 45-degree angle to x + y = 0.

The pair of points (-1,-1) and (3,7) lie on the same side of the line $3x - 8y - 7 = 0$ because when substituted into the equation, both points result in values with the same sign.

Option B (The lines are concurrent at the point (3/4, 1/2)) is correct because all lines satisfying the given condition pass through the point (3/4, 1/2).

Option C (Parallelogram) is correct because the pairs of lines are parallel, but the angle between them is not 90 degrees.

Option A ($\frac{\pi}{4}$) is correct because the angle between the lines is found using the slopes of the lines and the tangent formula.

The correct answer is D (None of these) because the solution requires finding the range of θ that satisfies the inequality derived from the point lying between the given lines, which doesn't match options A, B, or C.

Option A ($\left( {\sqrt 2 + \sqrt 6 ,\;\sqrt 2 - \sqrt 6 } \right)$) is correct because it represents one of the vertices of the equilateral triangle that lies on the given line.

Option A, $\left( {3 + \frac{2}{{\sqrt 3 }},3 + \frac{4}{{\sqrt 3 }}} \right)$, is correct because the triangle is equilateral, and the incenter coincides with the centroid, which can be calculated by averaging the coordinates.

Option A ($3x - 4y = 25$) is correct because it represents the equation of the line with the foot of the perpendicular from the origin at (3,-4).

Option D ($\frac{{3\mathrm{\pi }}}{{4}}$) is correct because the inclination is calculated using the midpoint formula and the arctangent function.