Chapter 10

Let \(A(2,-3)\) and \(B(-2,3)\) be vertices of a triangle \(A B C\). If the centroid of this triangle moves on the line \(2 x+3 y=1\), then the locus of the vertex \(C\) is the line [2004] (a) \(3 x-2 y=3\) (b) \(2 x-3 y=7\) (c) \(3 x+2 y=5\) (d) \(2 x+3 y=9\)

Locus of mid point of the portion between the axes of \(x \cos \alpha+y \sin \alpha=p\) whre \(p\) is constant is [2002] (a) \(x^{2}+y^{2}=\frac{4}{p^{2}}\) (b) \(x^{2}+y^{2}=4 p^{2}\) (c) \(\frac{1}{x^{2}}+\frac{1}{y^{2}}=\frac{2}{p^{2}}\) (d) \(\frac{1}{x^{2}}+\frac{1}{y^{2}}=\frac{4}{p^{2}}\)

Let \(L\) denote the line in the \(x y\)-plane with \(x\) and \(y\) intercepts as 3 and 1 respectively. Then the image of the point \((-1,-4)\) in this line is: (a) \(\left(\frac{11}{5}, \frac{28}{5}\right)\) (b) \(\left(\frac{29}{5}, \frac{8}{5}\right)\) (c) \(\left(\frac{8}{5}, \frac{29}{5}\right)\) (d) \(\left(\frac{29}{5}, \frac{11}{5}\right)\)

If the line, \(2 x-y+3=0\) is at a distance \(\frac{1}{\sqrt{5}}\) and \(\frac{2}{\sqrt{5}}\) from the lines \(4 x-2 y+\alpha=0\) and \(6 x-3 y+\beta=0\), respectively, then the sum of all possible value of \(\alpha\) and \(B\) is

The locus of the mid-points of the perpendiculars drawn from points on the line, \(x=2 y\) to the line \(x=y\) is: [Jan. 7, 2020 (II)] (a) \(2 x-3 y=0\) (b) \(5 x-7 y=0\) (c) \(3 x-2 y=0\) (d) \(7 x-5 y=0\)

A straight line \(L\) at a distance of 4 units from the origin makes positive intercepts on the coordinate axes and the perpendicular from the origin to this line makes an angle of \(60^{\circ}\) with the line \(x+y=0\). Then an equation of the line Lis: \(\quad\) [April 12, 2019 (II)] (a) \(x+\sqrt{3} y=8\) (b) \((\sqrt{3}+1) x+(\sqrt{3}-1) y=8 \sqrt{2}\) (c) \(\sqrt{3} x+y=8\) (d) None of these

Lines are drawn parallel to the line \(4 x-3 y+2=0\), at a distance \(\frac{3}{5}\) from the origin. Then which one of the following points lies on any of these lines ? [April 10, 2019 (II)] (a) \(\left(-\frac{1}{4}, \frac{2}{3}\right)\) (b) \(\left(\frac{1}{4},-\frac{1}{3}\right)\) (c) \(\left(\frac{1}{4}, \frac{1}{3}\right)\) (d) \(\left(-\frac{1}{4},-\frac{2}{3}\right)\)

If the two lines \(x+(a-1) y=1\) and \(2 x+a^{2} y=1(a \in \mathrm{R}-\\{0,1\\})\) are perpendicular, then the distance of their point of intersection from the origin is: [April09, 2019 (II)] (a) \(\sqrt{\frac{2}{5}}\) (b) \(\frac{2}{5}\) (c) \(\frac{2}{\sqrt{5}}\) (d) \(\frac{\sqrt{2}}{5}\)

A rectangle is inscribed in a circle with a diameter lying along the line \(3 y=x+7\). If the two adjacent vertices of the rectangle are \((-8,5)\) and \((6,5)\), then the area of the rectangle (in sq. units) is: [April 09, 2019 (II)] (a) 84 (b) 98 (c) 72 (d) 56

Suppose that the points \((h, k),(1,2)\) and \((-3,4)\) lie on the line \(\mathrm{L}_{1}\). If a line \(\mathrm{L}_{2}\) passing through the points \((h, k)\) and \((4,3)\) is perpendicular on \(\mathrm{L}_{1}\), then equals : [April08, 2019 (II)] (a) \(\frac{1}{3}\) (b) 0 (c) 3 (d) \(-\frac{1}{7}\)

If the straight line, \(2 x-3 y+17=0\) is perpendicular to the line passing through the points \((7,17)\) and \((15, \beta)\), then \(\beta\) equals: [Jan. \(12,2019(\mathrm{I})]\) (a) \(\frac{35}{3}\) (b) \(-5\) (c) \(-\frac{35}{3}\) (d) 5

Two sides of a parallelogram are along the lines, \(x+y=3\) and \(x-y+3=0\). If its diagonals intersect at \((2,4)\), then one of its vertex is: \(\quad\) [Jan. 10, 2019 (II)] (a) \((3,5)\) (b) \((2,1)\) (c) \((2,6)\) (d) \((3,6)\)

Consider the set of all lines \(p x+q y+r=0\) such that \(3 \mathrm{p}+2 \mathrm{q}+4 \mathrm{r}=0 .\) Which one of the following statements is true? \(\quad\) [Jan. 9,2019 (I)] (a) The lines are concurrent at the point \(\left(\frac{3}{4}, \frac{1}{2}\right)\). (b) Each line passes through the origin. (c) The lines are all parallel. (d) The lines are not concurrent.

Let the equations of two sides of a triangle be \(3 x-2 y+6=0\) and \(4 x+5 y-20=0 .\) If the orthocentre of this triangle is at \((1,1)\), then the equation of its third side is: \(\quad\) [Jan. 09, 2019 (II)] (a) \(122 y-26 x-1675=0\) (b) \(122 y+26 x+1675=0\) (c) \(26 x+61 y+1675=0\) (d) \(26 x-122 y-1675=0\)

The foot of the perpendicular drawn from the origin, on the line, \(3 x+y=\lambda(\lambda \neq 0)\) is \(P\). If the linemeets \(x\)-axisat \(A\) and \(y\)-axis at \(B\), then the ratio \(B P: P A\) is [Online April \(15, \mathbf{2 0 1 8}]\) (a) \(9: 1\) (b) \(1: 3\) (c) \(1: 9\) (d) \(3: 1\)

The sides of a rhombus \(A B C D\) are parallel to the lines, \(x-y+2=0\) and \(7 x-y+3=0\). If the diagonals of the rhombus intersect at \(P(1,2)\) and the vertex \(A\) (different from the origin) is on the \(y\)-axis, then the ordinate of \(A\) is [Online April 15, 2018] (a) 2 (b) \(\frac{7}{4}\) (c) \(\frac{7}{2}\) (d) \(\frac{5}{2}\)

Let \(\mathrm{a}, b, c\) and \(d\) be non-zero numbers. If the point of intersection of the lines \(4 a x+2 a y+c=0\) and \(5 b x+2 b y+d\) \(=0\) lies in the fourth quadrant and is equidistant from the two axes then (a) \(3 b c-2 a d=0\) (b) \(3 b c+2 a d=0\) (c) \(2 b c-3 a d=0\) (d) \(2 b c+3 a d=0\)

Let \(P S\) be the median of the triangle vertices \(P(2,2), Q(6,-1)\) and \(R(7,3)\). The equation of the line passing through \((1,-1)\) and parallel to PS is: (a) \(4 x+7 y+3=0\) (b) \(2 x-9 y-11=0\) (c) \(4 x-7 y-11=0\) (d) \(2 x+9 y+7=0\)

If a line \(L\) is perpendicular to the line \(5 x-y=1\), and the area of the triangle formed by the line \(L\) and the coordinate axes is 5, then the distance of line \(L\) from the line \(x+5 y=0\) is: \mathrm{\\{} O n l i n e ~ A p r i l ~ 1 9 , ~ 2 0 1 4 ] ~ (a) \(\frac{7}{\sqrt{5}}\) (b) \(\frac{5}{\sqrt{13}}\) (c) \(\frac{7}{\sqrt{13}}\) (d) \(\frac{5}{\sqrt{7}}\)

If the three distinct lines \(x+2 a y+a=0, x+3 b y+b=0\) and \(\mathrm{x}+4 \mathrm{ay}+\mathrm{a}=0\) are concurrent, then the point \((\mathrm{a}, \mathrm{b})\) lies on a: [Online April 12, 2014] (a) circle (b) hyperbola (c) straight line (d) parabola

Let \(\theta_{1}\) be the angle between two lines \(2 x+3 y+c_{1}=0\) and \(-x+5 y+c_{2}=0\) and \(\theta_{2}\) be the angle between two lines \(2 x+3 y+c_{1}=0\) and \(-x+5 y+c_{3}=0\), where \(c_{1}, c_{2}, c_{3}\) are any real numbers: Statement-1: If \(c_{2}\) and \(c_{3}\) are proportional, then \(\theta_{1}=\theta_{2}\). Statement-2: \(\theta_{1}=\theta_{2}\) for all \(c_{2}\) and \(c_{3}\). \mathrm{\\{} O n l i n e ~ A p r i l ~ 2 3 , ~ 2 0 1 3 ] ~ (a) Statement- 1 is true, Statement- 2 is true; Statement- 2 is a correct explanation of Statement-1. (b) Statement= 1 is true, Statement 2 is true; Statement- 2 is not a correct explanation of Statement-1. (c) Statement- 1 is false; Statement- 2 is true. (d) Statement- 1 is true; Statement- 2 is false.

If the three lines \(x-3 y=p, a x+2 y=q\) and \(a x+y=r\) form a right-angled triangle then : [Online April9, 2013] (a) \(a^{2}-9 a+18=0\) (b) \(a^{2}-6 a-12=0\) (c) \(a^{2}-6 a-18=0\) (d) \(a^{2}-9 a+12=0\)

Consider the straight lines \(L_{1}: x-y=1\) \(L_{2}: x+y=1\) \(L_{3}: 2 x+2 y=5\) \(L_{4}: 2 x-2 y=7\) The correct statement is \(\quad\) [Online May 26, 2012] (a) \(L_{1}\left\|L_{4}, L_{2}\right\| L_{3}, L_{1}\) intersect \(L_{4}\). (b) \(L_{1} \perp L_{2}, L_{1} \| L_{3}, L_{1}\) intersect \(L_{2}\). (c) \(L_{1} \perp L_{2}, L_{2} \| L_{3}, L_{1}\) intersect \(L_{4}\). (d) \(L_{1} \perp L_{2}, L_{1} \perp L_{3}, L_{2}\) intersect \(L_{4}\).

If \(a, b, c \in \mathrm{R}\) and 1 is a root of equation \(a x^{2}+b x+c=0\) then the curve \(y=4 a x^{2}+3 b x+2 c, a \neq 0\) intersect \(x\)-axis at [Online May 26, 2012] (a) two distinct points whose coordinates are always rational numbers (b) no point (c) exactly two distinct points (d) exactly one point

Let \(L\) be the line \(y=2 x\), in the two dimensional plane. [Online May \(19, \mathbf{2 0 1 2}]\) Statement \(1:\) The image of the point \((0,1)\) in \(L\) is the point \(\left(\frac{4}{5}, \frac{3}{5}\right)\) Statement 2: The points \((0,1)\) and \(\left(\frac{4}{5}, \frac{3}{5}\right)\) lie on opposite sides of the line \(\mathrm{L}\) and are at equal distance from it. (a) Statement 1 is true, Statement 2 is false. (b) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement \(1 .\) (c) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1 . (d) Statement 1 is false, Statement 2 is true.

If two vertices of a triangle are \((5,-1)\) and \((-2,3)\) and its orthocentre is at \((0,0)\), then the third vertex is [Online May 12, 2012] (a) \((4,-7)\) (b) \((-4,-7)\) (c) \((-4,7)\) (d) \((4,7)\)

The point of intersection of the lines \(\left(a^{3}+3\right) x+a y+a-3=0\) and \(\left(a^{5}+2\right) x+(a+2) y+2 a+3=0\) (a real) lies on the \(y\)-axis for |Online May 7, 2012] (a) no value of \(a\) (b) more than two values of \(a\) (c) exactly one value of \(a\) (d) exactly two values of \(a\)

The lines \(x+y=|a|\) and \(a x-y=1\) intersect each other in the first quadrant. Then the set of all possible values of \(a\) in the interval : (a) \((0, \infty)\) (b) \([1, \infty)\) (c) \((-1, \infty)\) (d) \((-1,1)\)

The lines \(L_{1}: y-x=0\) and \(L_{2}: 2 x+y=0\) intersect the line \(L_{3}: y+2=0\) at \(P\) and \(Q\) respectively. The bisector of the acute angle between \(L_{1}\) and \(L_{2}\) intersects \(L_{3}\) at \(R\). [2011] Statement-1: The ratio \(P R: R Q\) equals \(2 \sqrt{2}: \sqrt{5}\) Statement-2: In any triangle, bisector of an angle divides the triangle into two similar triangles. (a) Statement- 1 is true, Statement- 2 is true; Statement- 2 is not a correct explanation for Statement-1. (b) Statement- 1 is true, Statement- 2 is false. (c) Statement- 1 is false, Statement- 2 is true. (d) Statement- 1 is true, Statement- 2 is true; Statement- 2 is a correct explanation for Statement- 1 .

The lines \(p\left(p^{2}+1\right) x-y+q=0\) and \(\left(\mathrm{p}^{2}+1\right)^{2} \mathrm{x}+\left(\mathrm{p}^{2}+1\right) \mathrm{y}+2 \mathrm{q}=0\) are perpendicular to a common line for : \(\quad\) [2009] (a) exactly one values of \(\mathrm{p}\) (b) exactly two values of \(\mathrm{p}\) (c) more than two values of \(\mathrm{p}\) (d) no value of \(\mathrm{p}\)

The shortest distance between the line \(\mathrm{y}-\mathrm{x}=1\) and the curve \(x=y^{2}\) is: (a) \(\frac{2 \sqrt{3}}{8}\) (b) \(\frac{3 \sqrt{2}}{5}\) (c) \(\frac{\sqrt{3}}{4}\) (d) \(\frac{3 \sqrt{2}}{8}\)

The perpendicular bisector of the line segment joining \(\mathrm{P}(1,4)\) and \(\mathrm{Q}(\mathrm{k}, 3)\) has \(\mathrm{y}\)-intercept \(-4\). Then a possible value of \(\mathrm{k}\) is (a) 1 (b) 2 (c) \(-2\) (d) \(-4\)

Let \(\mathrm{P}=(-1,0), \mathrm{Q}=(0,0)\) and \(\mathrm{R}=(3,3 \sqrt{3})\) be three point. The equation of the bisector of the angle PQR is [2007] (a) \(\frac{\sqrt{3}}{2} x+y=0\) (b) \(x+\sqrt{3 y}=0\) (c) \(\sqrt{3} x+y=0\) (d) \(x+\frac{\sqrt{3}}{2} y=0\)

If \(x_{1}, x_{2}, x_{3}\) and \(y_{1}, y_{2}, y_{3}\) are both in G.P. with the same common ratio, then the points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\) and \(\left(x_{3}, y_{3}\right)\) (a) are vertices of a triangle (b) lie on a straight line (c) lie on an ellipse (d) lie on a circle.

A square of side a lies above the \(x\)-axis and has one vertex at the origin. The side passing through the origin makes an angle \(\alpha\left(0<\alpha<\frac{\pi}{4}\right)\) with the positive direction of \(x\)-axis. The equation of its diagonal not passing through the origin is [2003] (a) \(y(\cos \alpha+\sin \alpha)+x(\cos \alpha-\sin \alpha)=a\) (b) \(y(\cos \alpha-\sin \alpha)-x(\sin \alpha-\cos \alpha)=a\) (c) \(y(\cos \alpha+\sin \alpha)+x(\sin \alpha-\cos \alpha)=a\) (d) \(y(\cos \alpha+\sin \alpha)+x(\sin \alpha+\cos \alpha)=a\)

The equation \(y=\sin x \sin (x+2)-\sin ^{2}(x+1)\) represents a straight line lying in : [April 12, 2019 (I)] (a) second and third quadrants only (b) first, second and fourth quadrant (c) first, third and fourth quadrants (d) third and fourth quadrants only

If one of the lines of \(m y^{2}+\left(1-m^{2}\right) x y-m x^{2}=0\) is a bisector of the angle between the lines \(x y=0\), then \(\mathrm{m}\) is \(\quad\) [2007] (a) 1 (b) 2 (c) \(-1 / 2\) (d) \(-2\)

If one of the lines given by \(6 x^{2}-x y+4 c y^{2}=0\) is \(3 x+4 y=0\), then c equals [2004] (a) \(-3\) (b) 1 (c) 3 (d) 1

If the sum of the slopes of the lines given by \(x^{2}-2 c x y-7 y^{2}=0\) is four times their product \(c\) has the value (a) \(-2\) (b) \(-1\) (c) 2 (d) 1

If the pair of straight lines \(x^{2}-2 p x y-y^{2}=0\) and \(x^{2}-2 q x y-y^{2}=0\) be such that each pair bisects the angle between the other pair, then (a) \(p q=-1\) (b) \(p=q\) (c) \(p=-q\) (d) \(p q=1\).

The pair of lines represented by \(3 a x^{2}+5 x y+\left(a^{2}-2\right) y^{2}=0\) are perpendicular to each other for (a) two values of \(a\) (b) \(\forall a\) (c) for one value of \(a\) (d) for no values of \(a\)

If the pair of lines \(a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c=0\) intersect on the \(y\)-axis then \([\mathbf{2 0 0 2}]\) (a) \(2 f g h=b g^{2}+c h^{2}\) (b) \(b g^{2} \neq c h^{2}\) (c) \(a b c=2 f g h\) (d) none of these