Chapter 10

A triangle \(A B C\) lying in the first quadrant has two vertices as \(A(1,2)\) and \(B(3,1)\). If \(\angle B A C=90^{\circ}\), andar \((\triangle A B C)=5 \sqrt{5}\) sq. units, then the abscissa of the vertex \(C\) is: (a) \(1+\sqrt{5}\) (b) \(1+2 \sqrt{5}\) (c) \(2+\sqrt{5}\) (d) \(2 \sqrt{5}-1\)

If the perpendicular bisector of the line segment joining the points \(P(1,4)\) and \(Q(k, 3)\) has \(y\)-intercept equal to \(-4\), then a value of \(k\) is: \(\quad\) [Sep. 04, 2020 (II)] (a) \(-2\) (b) \(-4\) (c) \(\sqrt{14}\) (d) \(\sqrt{15}\)

If a \(\triangle A B C\) has vertices \(A(-1,7), B(-7,1)\) and \(C(5,-5)\), then its orthocentre has coordinates: \(\quad\) [Sep.03, 2020 (II)] (a) \(\left(-\frac{3}{5}, \frac{3}{5}\right)\) (b) \((-3,3)\) (c) \(\left(\frac{3}{5},-\frac{3}{5}\right)\) (d) \((3,-3)\)

Let \(A(1,0), B(6,2)\) and \(C\left(\frac{3}{2}, 6\right)\) be the vertices of a triangle \(A B C\). If \(P\) is a point inside the triangle \(A B C\) such that the triangles \(A P C, A P B\) and \(B P C\) have equal areas, then the length of the line segment \(P Q\), where \(Q\) is the point \(\left(-\frac{7}{6},-\frac{1}{3}\right)\), is

A triangle has a vertex at \((1,2)\) and the mid points of the two sides through it are \((-1,1)\) and \((2,3)\). Then the centroid of this triangle is : \(\quad\) [April 12, 2019 (II)] (a) \(\left(1, \frac{7}{3}\right)\) (b) \(\left(\frac{1}{3}, 2\right)\) (c) \(\left(\frac{1}{3}, 1\right)\) (d) \(\left(\frac{1}{3}, \frac{5}{3}\right)\)

Let \(\mathrm{O}(0,0)\) and \(\mathrm{A}(0,1)\) be two fixed points. Then the locus of a point \(\mathrm{P}\) such that the perimeter of \(\triangle \mathrm{AOP}\) is 4, is : [April 8,2019 (I)] (a) \(8 x^{2}-9 y^{2}+9 y=18\) (b) \(9 x^{2}-8 y^{2}+8 y=16\) (c) \(9 x^{2}+8 y^{2}-8 y=16\) (d) \(8 x^{2}+9 y^{2}-9 y=18\)

Two vertices of a triangle are \((0,2)\) and \((4,3)\). If its orthocentre is at the origin, then its third vertex lies in which quadrant? \(\quad\) [Jan. 10, 2019 (II)] (a) third (b) second (c) first (d) fourth

Let the orthocentre and centroid of a triangle be \(\mathrm{A}(-3,5)\) and \(\mathrm{B}(3,3)\) respectively. If \(\mathrm{C}\) is the circumcentre of this triangle, then the radius of the circle having line segment AC as diameter, is : (a) \(2 \sqrt{10}\) (b) \(3 \sqrt{\frac{5}{2}}\) (c) \(\frac{3 \sqrt{5}}{2}\) (d) \(\sqrt{10}\)

A square, of each side 2 , lies above the \(x\)-axis and has one vertex at the origin. If one of the sides passing through the origin makes an angle \(30^{\circ}\) with the positive direction of the \(x\)-axis, then the sum of the \(x\)-coordinates of the vertices of the square is : \(\quad\) [Online April 9, 2017] (b) \(2 \sqrt{3}-2\) (c) \(\sqrt{3}-2\) (d) \(\sqrt{3}-1\) (a) \(2 \sqrt{3}-1\)

A ray of light is incident along a line which meets another line, \(7 x-y+1=0\), at the point \((0,1)\). The ray is then reflected from this point along the line, \(y+2 x=1\). Then the equation of the line of incidence of the ray of light is : [Online April \(\mathbf{1 0}\), 2016] (a) \(41 \mathrm{x}-25 \mathrm{y}+25=0\) (b) \(41 x+25 y-25=0\) (c) \(41 x-38 y+38=0\) (d) \(41 x+38 y-38=0\)

Let \(L\) be the line passing through the point \(P(1,2)\) such that its intercepted segment between the co-ordinate axes is bisected at \(\mathrm{P}\). If \(\mathrm{L}_{1}\) is the line perpendicular to \(\mathrm{L}\) and passing through the point \((-2,1)\), then the point of intersection of \(L\) and \(L_{1}\) is: \(\quad\) [Online April 10,2015\(]\) (a) \(\left(\frac{4}{5}, \frac{12}{5}\right)\) (b) \(\left(\frac{3}{5}, \frac{23}{10}\right)\) (c) \(\left(\frac{11}{20}, \frac{29}{10}\right)\) (d) \(\left(\frac{3}{10}, \frac{17}{5}\right)\)

The points \(\left(0, \frac{8}{3}\right),(1,3)\) and \((82,30)\) : [Online April 10, 2015] (a) form an acute angled triangle. (b) form a right angled triangle. (c) lie on a straight line. (d) form an obtuse angled triangle.

The \(x\)-coordinate of the incentre of the triangle that has the coordinates of mid points of its sides as \((0,1)(1,1)\) and \((1,0)\) is: [2013] (a) \(2+\sqrt{2}\) (b) \(2-\sqrt{2}\) (c) \(1+\sqrt{2}\) (d) \(1-\sqrt{2}\)

A light ray emerging from the point source placed at \(\mathrm{P}(1,3)\) is reflected at a point \(\mathrm{Q}\) in the axis of \(x\). If the reflected ray passes through the point \(\mathrm{R}(6,7)\), then the abscissa of \(\mathrm{Q}\) is: \mathrm{\\{} O n l i n e ~ A p r i l 9 , ~ \(\mathbf{2 0 1 3}]\) (a) 1 (b) 3 (c) \(\frac{7}{2}\) (d) \(\frac{5}{2}\)

Let \(\mathrm{A}(\mathrm{h}, \mathrm{k}), \mathrm{B}(1,1)\) and \(\mathrm{C}(2,1)\) be the vertices of a right angled triangle with \(\mathrm{AC}\) as its hypotenuse. If the area of the triangle is 1 square unit, then the set of values which \(\mathrm{k}\) ' can take is given by [2007] (a) \(\\{-1,3\\}\) (b) \(\\{-3,-2\\}\) (c) \(\\{1,3\\}\) (d) \(\\{0,2\\}\)

If a vertex of a triangle is \((1,1)\) and the mid points of two sides through this vertex are \((-1,2)\) and \((3,2)\) then the centroid of the triangle is (a) \(\left(-1, \frac{7}{3}\right)\) (c) \(\left(1, \frac{7}{3}\right)\) (b) \(\left(\frac{-1}{3}, \frac{7}{3}\right)\) (d) \(\left(\frac{1}{3}, \frac{7}{3}\right)\)

Locus of centroid of the triangle whose vertices are \((a \cos t, a \sin t),(b \sin t,-b \cos t)\) and \((1,0)\), where \(t\) is a parameter, is \([2003]\) (a) \((3 x+1)^{2}+(3 y)^{2}=a^{2}-b^{2}\) (b) \((3 x-1)^{2}+(3 y)^{2}=a^{2}-b^{2}\) (c) \((3 x-1)^{2}+(3 y)^{2}=a^{2}+b^{2}\) (d) \((3 x+1)^{2}+(3 y)^{2}=a^{2}+b^{2}\)

A triangle with vertices \((4,0),(-1,-1),(3,5)\) is (a) isosceles and right angled (b) isosceles but not right angled (c) right angled but not isosceles (d) neither right angled nor isosceles

Let \(f: \mathrm{R} \rightarrow \mathrm{R}\) be defined as \(f(x)=\left\\{\begin{array}{cc}x^{5} \sin \left(\frac{1}{x}\right)+5 x^{2}, & x<0 \\ 0, & x=0 \\ x^{5} \cos \left(\frac{1}{x}\right)+\lambda x^{2}, & x>0\end{array}\right.\) The value of \(\lambda\) for which \(f^{\prime \prime}(0)\) exists, is

Let \(\mathrm{C}\) be the centroid of the triangle with vertices \((3,-1)\), \((1,3)\) and \((2,4) .\) Let \(\mathrm{P}\) be the point of intersection of the lines \(x+3 y-1=0\) and \(3 x-y+1=0 .\) Then the line passing through the points \(\mathrm{C}\) and \(\mathrm{P}\) also passes through the point: [Jan. \(9,2020(\mathrm{I})]\) (a) \((-9,-6)\) (b) \((9,7)\) (c) \((7,6)\) (d) \((-9,-7)\)

Slope of a line passing through \(\mathrm{P}(2,3)\) and intersecting the line \(x+y=7\) at a distance of 4 units from \(P\), is: [April 9, 2019 (I)] (a) \(\frac{1-\sqrt{5}}{1+\sqrt{5}}\) (b) \(\frac{1-\sqrt{7}}{1+\sqrt{7}}\) (c) \(\frac{\sqrt{7}-1}{\sqrt{7}+1}\) (d) \(\frac{\sqrt{5}-1}{\sqrt{5}+1}\)

A point on the straight line, \(3 x+5 y=15\) which is equidistant from the coordinate axes will lie only in : [April 8, 2019 (I)] (a) \(4^{\text {th }}\) quadrant (b) \(1^{\text {st }}\) quadrant (c) \(1^{\text {st }}\) and \(2^{\text {wd }}\) quadrants (d) \(1^{\text {st }}, 2^{\text {nd }}\) and \(4^{\text {th }}\) quadrants

Two vertical poles of heights, \(20 \mathrm{~m}\) and \(80 \mathrm{~m}\) stand apart on a horizontal plane. The height (in meters) of the point of intersection of the lines joining the top of each pole to the foot of the other, from this horizontal plane is : [April08, 2019 (II)] (a) 15 (b) 18 (c) 12 (d) 16

If a straight line passing through the point \(\mathrm{P}(-3,4)\) is such that its intercepted portion between the coordinate axes is bisected at \(P\), then its equation is: \(\quad\) [Jan. 12, 2019 (II)] (a) \(3 x-4 y+25=0\) (b) \(4 x-3 y+24=0\) (c) \(x-y+7=0\) (d) \(4 x+3 y=0\)

If in a parallelogram \(\mathrm{ABDC}\), the coordinates of \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\) are respectively \((1,2),(3,4)\) and \((2,5)\), then the equation of the diagonal \(\mathrm{AD}\) is : [Jan. 11, 2019 (II)] (a) \(5 x-3 y+1=0\) (b) \(5 x+3 y-11=0\) (c) \(3 x-5 y+7=0\) (d) \(3 x+5 y-13=0\)

A point \(\mathrm{P}\) moves on the line \(2 x-3 y+4=0\). If \(\mathrm{Q}(1,4)\) and \(\mathrm{R}(3,-2)\) are fixed points, then the locus of the centroid of \(\Delta P Q R\) is a line: (a) with slope \(\frac{3}{2}\) (b) parallel to \(x\)-axis (c) with slope \(\frac{2}{3}\) (d) parallel to \(y\)-axis

If the line \(3 x+4 y-24=0\) intersects the \(x\)-axis at the point A and the \(y\)-axis at the point \(B\), then the incentre of the triangle OAB, where \(\mathrm{O}\) is the origin, is: [Jan. \(\mathbf{1 0}, \mathbf{2 0 1 9}\) (I)] (a) \((3,4)\) (b) \((2,2)\) (c) \((4,3)\) (d) \((4,4)\)

A straight line through a fixed point \((2,3)\) intersects the coordinate axes at distinct points \(P\) and \(Q\). If \(O\) is the origin and the rectangle OPRQ is completed, then the locus of \(\mathrm{R}\) is: \(\quad \mathbf{[ 2 0 1 8}]\) (a) \(2 x+3 y=x y\) (b) \(3 x+2 y=x y\) (c) \(3 x+2 y=6 x y\) (d) \(3 x+2 y=6\)

In a triangle \(A B C\), coordianates of \(A\) are \((1,2)\) and the equations of the medians through \(B\) and \(C\) are \(x+y=5\) and \(x=4\) respectively. Then area of \(\Delta A B C\) (in sq. units) is [Online April 15, 2018] (a) 5 (b) 9 (c) 12 (d) 4

Two sides of a rhombus are along the lines, \(x-y+1=0\) and \(7 x-y-5=0\). If its diagonals intersect at \((-1,-2)\), then which one of the following is a vertex of this rhombus? [2016] (a) \(\left(\frac{1}{3},-\frac{8}{3}\right)\) (b) \(\left(-\frac{10}{3},-\frac{7}{3}\right)\) (c) \((-3,-9)\) (d) \((-3,-8)\)

A straight line through origin \(\mathrm{O}\) meets the lines \(3 \mathrm{y}=10-4 \mathrm{x}\) and \(8 x+6 y+5=0\) at points A and B respectively. Then \(O\) divides the segment \(\mathrm{AB}\) in the ratio: [Online April 10, 2016] (a) \(2: 3\) (b) \(1: 2\) (c) \(4: 1\) (d) \(3: 4\)

If a variable line drawn through the intersection of the lines \(\frac{x}{3}+\frac{y}{4}=1\) and \(\frac{x}{4}+\frac{y}{3}=1\), meets the coordinate axes at \(\mathrm{A}\) and \(\mathrm{B},(\mathrm{A} \neq \mathrm{B})\), then the locus of the midpoint of \(\mathrm{AB}\) is: \(\quad\) [Online April 9, 2016] (a) \(7 x y=6(x+y)\) (b) \(4(x+y)^{2}-28(x+y)+49=0\) (c) \(6 x y=7(x+y)\) (d) \(14(x+y)^{2}-97(x+y)+168=0\)

The point \((2,1)\) is translated parallel to the line \(L: x-y=4\) by \(2 \sqrt{3}\) units. If the new points \(\mathrm{Q}\) lies in the third quadrant, then the equation of the line passing through \(\mathrm{Q}\) and perpendicular to \(L\) is : [Online April 9, 2016] (a) \(x+y=2-\sqrt{6}\) (b) \(2 x+2 y=1-\sqrt{6}\) (c) \(x+y=3-3 \sqrt{6}\) (d) \(x+y=3-2 \sqrt{6}\)

A straight line \(L\) through the point \((3,-2)\) is inclined at an angle of \(60^{\circ}\) to the line \(\sqrt{3} x+y=1 .\) If \(L\) also intersects the \(x\)-axis, then the equation of \(L\) is : [Online April 11, 2015] (a) \(y+\sqrt{3} x+2-3 \sqrt{3}=0\) (b) \(\sqrt{3} \mathrm{y}+\mathrm{x}-3+2 \sqrt{3}=0\) (c) \(y-\sqrt{3} x+2+3 \sqrt{3}=0\) (d) \(\sqrt{3} y-x+3+2 \sqrt{3}=0\)

The circumcentre of a triangle lies at the origin and its centroid is the mid point of the line segment joining the points \(\left(a^{2}+1, a^{2}+1\right)\) and \((2 a,-2 a), a \neq 0 .\) Then for any a, the orthocentre of this triangle lies on the line: \mathrm{\\{} O n l i n e ~ A p r i l ~ 1 9 , ~ 2 0 1 4 ] ~ (a) \(y-2 a x=0\) (b) \(y-\left(a^{2}+1\right) x=0\) (c) \(y+x=0\) (d) \((a-1)^{2} x-(a+1)^{2} y=0\)

If a line intercepted between the coordinate axes is trisected at a point \(\mathrm{A}(4,3)\), which is nearer to \(\mathrm{x}\)-axis, then its equation is: \([\) Online April 12, 2014] (a) \(4 x-3 y=7\) (b) \(3 x+2 y=18\) (c) \(3 x+8 y=36\) (d) \(x+3 y=13\)

Given three points \(\mathrm{P}, \mathrm{Q}, \mathrm{R}\) with \(\mathrm{P}(5,3)\) and \(\mathrm{R}\) lies on the \(x\)-axis. If equation of \(R Q\) is \(x-2 y=2\) and \(P Q\) is parallel to the \(x\)-axis, then the centroid of \(\Delta\) PQR lies on the line: [Online April 9, 2014] (a) \(2 x+y-9=0\) (b) \(x-2 y+1=0\) (c) \(5 x-2 y=0\) (d) \(2 x-5 y=0\)

A ray of light along \(x+\sqrt{3} y=\sqrt{3}\) gets reflected upon reaching \(x\)-axis, the equation of the reflected ray is [2013] (a) \(y=x+\sqrt{3}\) (b) \(\sqrt{3} y=x-\sqrt{3}\) (c) \(y=\sqrt{3} x-\sqrt{3}\) (d) \(\sqrt{3} y=x-1\)

Let \(\mathrm{A}(-3,2)\) and \(\mathrm{B}(-2,1)\) be the vertices of a triangle \(\mathrm{ABC}\). If the centroid of this triangle lies on the line \(3 x+4 y+2=0\), then the vertex \(C\) lies on the line : [Online April 25, 2013] (a) \(4 x+3 y+5=0\) (b) \(3 x+4 y+3=0\) (c) \(4 x+3 y+3=0\) (d) \(3 x+4 y+5=0\)

If the extremities of the base of an isosceles triangle are the points \((2 a, 0)\) and \((0, a)\) and the equation of one of the sides is \(x=2 a\), then the area of the triangle, in square units, is: [Online April 23, 2013] (a) \(\frac{5}{4} a^{2}\) (b) \(\frac{5}{2} a^{2}\) (c) \(\frac{25 a^{2}}{4}\) (d) \(5 a^{2}\)

If the \(x\)-intercept of some line \(L\) is double as that of the line, \(3 x+4 y=12\) and the \(y\)-intercept of \(L\) is half as that of the same line, then the slope of \(L\) is : [Online April 22, 2013] (a) \(-3\) (b) \(-3 / 8\) (c) \(-3 / 2\) (d) \(-3 / 16\)

If the line \(2 x+y=k\) passes through the point which divides the line segment joining the points \((1,1)\) and \((2,4)\) in the ratio \(3: 2\), then \(k\) equals : (a) \(\frac{29}{5}\) (b) 5 (c) 6 (d) \(\frac{11}{5}\)

The line parallel to \(x\)-axis and passing through the point of intersection of lines \(a x+2 b y+3 b=0\) and \(b x-2 a y-3 a=0\), where \((a, b) \neq(0,0)\) is \(\quad\) [Online May 26, 2012] (a) above \(x\)-axis at a distance \(2 / 3\) from it (b) above \(x\)-axis at a distance \(3 / 2\) from it (c) below \(x\)-axis at a distance \(3 / 2\) from it (d) below \(x\)-axis at a distance \(2 / 3\) from it

If the point \((1, a)\) lies between the straight lines \(x+y=1\) and \(2(x+y)=3\) then a lies in interval \mathrm{\\{} O n l i n e ~ M a y ~ 1 2 , ~ 2 0 1 2 ] ~ (b) \(\left(1, \frac{3}{2}\right)\) (a) \(\left(\frac{3}{2}, \infty\right)\) (c) \((-\infty, 0)\) (d) \(\left(0, \frac{1}{2}\right)\)

If the straight lines \(x+3 y=4,3 x+y=4\) and \(x+y=0\) form a triangle, then the triangle is \(\quad\) Online May 7, 2012] (a) scalene (b) equilateral triangle (c) isosceles \(\mathbf{5 2}\) (d) right angled isosceles

If \(\mathrm{A}(2,-3)\) and \(\mathrm{B}(-2,1)\) are two vertices of a triangle and third vertex moves on the line \(2 x+3 y=9\), then the locus of the centroid of the triangle is : (a) \(x-y=1\) (b) \(2 x+3 y=1\) (c) \(2 x+3 y=3\) (d) \(2 x-3 y=1\)

If \(\left(a, a^{2}\right)\) falls inside the angle made by the lines \(y=\frac{x}{2}\), \(x>0\) and \(y=3 x, x>0\), then a belong to [2006] (a) \(\left(0, \frac{1}{2}\right)\) (b) \((3, \infty)\) (c) \(\left(\frac{1}{2}, 3\right)\) (d) \(\left(-3,-\frac{1}{2}\right)\)

A straight line through the point \(\mathrm{A}(3,4)\) is such that its intercept between the axes is bisected at \(\mathrm{A}\). Its equation is (a) \(x+y=7\) (b) \(3 x-4 y+7=0\) (c) \(4 x+3 y=24\) (d) \(3 x+4 y=25\)

The line parallel to the \(x\)-axis and passing through the intersection of the lines \(a x+2 b y+3 b=0\) and \(b x-2 a y-3 a=0\), where \((a, b) \neq(0,0)\) is \(\quad\) [2005] (a) below the \(x\)-axis at a distance of \(\frac{3}{2}\) from it (b) below the \(x\)-axis at a distance of \(\frac{2}{3}\) from it (c) above the \(x\)-axis at a distance of \(\frac{3}{2}\) from it (d) above the \(x\)-axis at a distance of \(\frac{2}{3}\) from it

The equation of the straight line passing through the point \((4,3)\) and making intercepts on the co-ordinate axes whose sum is \(-1\) is (a) \(\frac{x}{2}-\frac{y}{3}=1\) and \(\frac{x}{-2}+\frac{y}{1}=1\) (b) \(\frac{x}{2}-\frac{y}{3}=-1\) and \(\frac{x}{-2}+\frac{y}{1}=-1\) (c) \(\frac{x}{2}+\frac{y}{3}=1\) and \(\frac{x}{2}+\frac{y}{1}=1\) (d) \(\frac{x}{2}+\frac{y}{3}=-1\) and \(\frac{x}{-2}+\frac{y}{1}=-1\)