Straight lines practice · Mixed difficulty

Option C ($c\cos \mathrm{\alpha }$) is correct because it represents the length of the perpendicular from the given point to the line, calculated using the perpendicular distance formula.

Option A ($ab' = a'b$) is correct because if the points are collinear, the determinant of the matrix formed by the coordinates must be zero, leading to the given relationship.

Option D ($bx - ay = 0$) is correct because it represents the locus of points equidistant from the two given points, derived by equating the squared distances.

Option A (statement (i) has more solution than statement (ii) for $\theta$) is correct because statement (i) yields more solutions for $\theta$ than statement (ii) due to the different trigonometric forms.

The correct answer is C (6) because the value of k is found by substituting the coordinates of the point dividing the line segment in the given ratio into the equation of the line.

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.

Option B is correct. If the arithmetic mean of reciprocals of intercepts of the line $y=mx+c$ on coordinate axes is $1/4$, then this line will pass through the point $(2,2)$.

Option B, x-y+2=0, is the correct bisector of the angle between the two lines that contains the point (2,1).

Option A ($\pm \frac{\pi }{2}$) is incorrect. The area of the parallelogram is minimized when the angle between the lines is maximized, which occurs at $\pm \frac{\pi }{2}$. The question asks when the area is LEAST, not maximum. The correct answer should be when the area is largest, which is when the angle is smallest.

Option B ($18 x+12 y+11=0$) is correct because it represents the line equidistant from the two given parallel lines.

Option B (a family of concurrent lines) is correct because when a, b, c are in A.P., the line equation can be rearranged to represent lines passing through a fixed point.

The correct answer is D (8) because it is calculated using the formula for the distance between two parallel lines.

The correct answer is D ((7,7)) because it involves finding the equation of the line L using the midpoint and slope of PQ, then finding the image of R in that line.

The correct answer is D ($4+6 \sqrt{3}$) because it represents the distance calculated using parametric coordinates and the given line equations.

The correct answer is D (exactly one point) because by substituting the first equation into the second, we get a linear equation, which intersects the x-axis at only one point.

The correct answer is A ($y = \tan \left( {\theta - 30^\circ } \right)\left( {x - 4 - 3\sqrt 5 } \right)$) because it represents the equation of the translated and rotated line, considering the shift and the clockwise rotation about the new x-intercept.

Option C ($x + 2y = 2$) is correct because it bisects the join of the points and has the specified intercepts ratio.

Option D ($4: 9$) is correct because it represents the ratio in which the line divides the line segment, calculated using the section formula.

Option B (3:4 internally) is correct because the ratio is calculated by substituting the points into the line equation and taking the negative ratio.

Option C ($x-y-1=0$) is correct because it represents a line equally inclined to the axes and equidistant from the given points, which is found by solving for the constant term in the line equation.

Option B ($\mathrm{\alpha \beta }\left( {x + y} \right) = 2xy\left( {\mathrm{\alpha } + \mathrm{\beta }} \right)$) is correct because it represents the locus of the midpoint of AB based on the given conditions.

Option A (1/2, 1/3) is correct because it represents the fixed point through which the family of lines passes, derived by setting up and solving simultaneous equations.

Option A ($x(\sqrt{{13}}+2 \sqrt{{2}})+y (\sqrt{{13}}-3 \sqrt{{2}}) +(\sqrt{{13}}-5 \sqrt{{2}})=0$) is correct because it represents the angle bisector that contains the given point (10, -20).

Option D (None of these) is correct because the calculated equation of the diagonal through B does not match any of the given options.

Option A ($\left( {\frac{7}{3},\frac{5}{3}} \right)$) is correct because the centroid divides the line segment joining the orthocenter and circumcenter in the ratio 2:1.

Option C, $\frac{{2(12 \sqrt{{3}}-8)}}{{6 \sqrt{{3}}+3}}$, is the correct ratio of PQ to PR, calculated by finding the intersection points of the line with the given lines and using the distance formula.

Option B ($\frac{{8}}{{13}}$) is correct because it is the correct distance calculated using the point-to-line distance formula.

Option B (1,1) is correct because the incenter of the triangle formed by the given lines is calculated using the formula for the incenter of a right-angled triangle.

Option A (A circle) is correct because the locus of a point P such that angle APB is a right angle is a circle with AB as the diameter.

Option B ($\left(-\frac{1}{2}, \frac{3}{2}\right)$) is correct because it represents the coordinates of the point on the line $x+y=1$ that is closest to (2,4).