Chapter 18

In oblique coordinates, the equation \(y=m x+c\) represents a straight line which is inclined at an angle $$ \tan ^{-1}\left(\frac{m \sin w}{1+m \cos w}\right) $$ to the \(x\)-axis, where \(w\) is the angle between the axes. If \(\theta\) be the angle between two lines \(y=m_{1} x+c_{1}\) and \(y=m_{2} x\) \(+c_{2}, w\) be the angle between the axes, then $$ \tan \theta=\frac{\left(m_{1}-m_{2}\right) \sin w}{1+\left(m_{1}+m_{2}\right) \cos w+m_{1} m_{2}} $$ The two given lines are parallel if \(m_{1}=m_{2}\). The two lines are perpendicular if \(1+\left(m_{1}+m_{2}\right) \cos w+\) \(m_{1} m_{2}=0\) If the straight lines \(y=m_{1} x+c_{1}\) and \(y=m_{2} x+c_{2}\) make equal angles with the axis of \(x\) and be not parallel to one another, then \(m_{1}+m_{2}+k m_{1} m_{2} \cos w=0\) where \(k=\) (A) 1 (B) 2 (C) \(-1\) (D) \(-2\)

In oblique coordinates, the equation \(y=m x+c\) represents a straight line which is inclined at an angle $$ \tan ^{-1}\left(\frac{m \sin w}{1+m \cos w}\right) $$ to the \(x\)-axis, where \(w\) is the angle between the axes. If \(\theta\) be the angle between two lines \(y=m_{1} x+c_{1}\) and \(y=m_{2} x\) \(+c_{2}, w\) be the angle between the axes, then $$ \tan \theta=\frac{\left(m_{1}-m_{2}\right) \sin w}{1+\left(m_{1}+m_{2}\right) \cos w+m_{1} m_{2}} $$ The two given lines are parallel if \(m_{1}=m_{2}\). The two lines are perpendicular if \(1+\left(m_{1}+m_{2}\right) \cos w+\) \(m_{1} m_{2}=0\) The axes being inclined at an angle of \(30^{\circ}\), the slope of the line which passes through the point \((-2,3)\) and is perpendicular to the straight line \(y+3 x=6\) is (A) \(\frac{3 \sqrt{3}-2}{\sqrt{3}-6}\) (B) \(\frac{3 \sqrt{3}+2}{\sqrt{3}-6}\) (C) \(\frac{3 \sqrt{3}-2}{\sqrt{3}+6}\) (D) none of these

In oblique coordinates, the equation \(y=m x+c\) represents a straight line which is inclined at an angle $$ \tan ^{-1}\left(\frac{m \sin w}{1+m \cos w}\right) $$ to the \(x\)-axis, where \(w\) is the angle between the axes. If \(\theta\) be the angle between two lines \(y=m_{1} x+c_{1}\) and \(y=m_{2} x\) \(+c_{2}, w\) be the angle between the axes, then $$ \tan \theta=\frac{\left(m_{1}-m_{2}\right) \sin w}{1+\left(m_{1}+m_{2}\right) \cos w+m_{1} m_{2}} $$ The two given lines are parallel if \(m_{1}=m_{2}\). The two lines are perpendicular if \(1+\left(m_{1}+m_{2}\right) \cos w+\) \(m_{1} m_{2}=0\) If \(y=x \tan \frac{11 \pi}{24}\) and \(y=x \tan \frac{19 \pi}{24}\) represent two straight lines at right angles, then the angle between the axes is (A) \(\frac{\pi}{6}\) (B) \(\frac{\pi}{4}\) (C) \(\frac{\pi}{3}\) (D) \(\frac{\pi}{2}\)

In oblique coordinates, the equation \(y=m x+c\) represents a straight line which is inclined at an angle $$ \tan ^{-1}\left(\frac{m \sin w}{1+m \cos w}\right) $$ to the \(x\)-axis, where \(w\) is the angle between the axes. If \(\theta\) be the angle between two lines \(y=m_{1} x+c_{1}\) and \(y=m_{2} x\) \(+c_{2}, w\) be the angle between the axes, then $$ \tan \theta=\frac{\left(m_{1}-m_{2}\right) \sin w}{1+\left(m_{1}+m_{2}\right) \cos w+m_{1} m_{2}} $$ The two given lines are parallel if \(m_{1}=m_{2}\). The two lines are perpendicular if \(1+\left(m_{1}+m_{2}\right) \cos w+\) \(m_{1} m_{2}=0\) The axes being inclined at an angle of \(120^{\circ}\), the tangent of the angle between the two straight lines \(8 x+7 y\) \(=1\) and \(28 x-73 y=101\) is \(\tan ^{-1} \theta\), where \(\theta=\) (A) \(\frac{30 \sqrt{3}}{37}\) (B) \(\frac{15 \sqrt{3}}{37}\) (C) \(\frac{7 \sqrt{3}}{37}\) (D) none of these

Column-I Column-II I. The diagonals of the parallelogram (A) \(\frac{5 \pi}{12}\) whose sides are \(l x+m y+n=0, l x+\) \(m y+n=0, m x+l y+n=0, m x+l y+\) \(n^{\prime}=0\) include an angle II. The line \(x+y=2\) turns about the point (B) \(\frac{\pi}{12}\) on it, whose ordinate is equal to abscissa, through an angle \(\theta\) in the clockwise direction so that its equation becomes \(y\) \(=2 x-1\). Then, the value of the angle \(\theta\) is III. The larger of the two angles made with the \(x\)-axis of a straight line drawn (C) \(\frac{\pi}{2}\) through \((1,2)\) so that it intersects \(x+y\) \(=4\) at a point distant \(\frac{\sqrt{6}}{3}\) from \((1,2)\) is (D) \(\tan ^{-1} 3\)

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

The equation of the directrix of the parabola \(y^{2}+4 y+\) \(4 x+2=0\) is: (A) \(x=-1\) (B) \(x=1\) (C) \(x=-\frac{3}{2}\) (D) \(x=\frac{3}{2}\)

The incentre of the triangle with vertices \((1, \sqrt{3}),(0,\), 0) and \((2,0)\) is: (A) \(\left(1, \frac{\sqrt{3}}{2}\right)\) (B) \(\left(\frac{2}{3}, \frac{1}{\sqrt{3}}\right)\) (C) \(\left(\frac{2}{3}, \frac{\sqrt{3}}{2}\right)\) (D) \(\left(1, \frac{1}{\sqrt{3}}\right)\)

Three straight lines \(2 x+11 y-5=0,24 x+7 y-20=\) 0 and \(4 x-3 y-2=0\) : (A) form a triangle (B) are only concurrent (C) are concurrent with one line bisecting the angle between the other two (D) none of the above

A straight line through the point \((2,2)\) intersects the lines \(\sqrt{3} x+y=0\) and \(\sqrt{3} x-y=0\) at the points \(A\) and \(B\). The equation to the line \(A B\) so that the triangle \(O A B\) is equilateral, is: (A) \(x-2=0\) (B) \(y-2=0\) (C) \(x+y-4=0\) (D) none of these

If the equation of the locus of a point equidistant from the points \(\left(a_{1}, b_{1}\right)\) and \(\left(a_{2}, b_{2}\right)\) is \(\left(a_{1}-a_{2}\right) x+\left(b_{1}-b_{2}\right) y\) \(+c=0\), then the value of \(^{\circ} c^{\prime}\) is \(\quad\) (A) \(\frac{1}{2}\left(a_{2}^{2}+b_{2}^{2}-a_{1}^{2}-b_{1}^{2}\right)\) (B) \(a_{1}^{2}+a_{2}^{2}-b_{1}^{2}-b_{2}^{2}\) (C) \(\frac{1}{2}\left(a_{1}^{2}+a_{2}^{2}-b_{1}^{2}-b_{2}^{2}\right)\) (D) \(\sqrt{a_{1}^{2}+b_{1}^{2}-a_{2}^{2}-b_{2}^{2}}\)

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 (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) \((3 x+1)^{2}+(3 y)^{2}=a^{2}-b^{2}\)

Let \(A(2,-3)\) and \(B(-2,1)\) 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 (A) \(2 x+3 y=9\) (B) \(2 x-3 y=7\) (C) \(3 x+2 y=5\) (D) \(3 x-2 y=3\)

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 \(\frac{x}{-2}+\frac{y}{1}=1\)

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, then \(c\) has the value (A) 1 (B) \(-1\) (C) 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 (A) 1 (B) \(-1\) (C) 3 (D) \(-3\)

Let \(P\) be the point \((1,0)\) and \(Q\) a point on the locus \(y^{2}\) \(=8 x\). The locus of mid-point of \(P Q\) is (A) \(y^{2}-4 x+2=0\) (B) \(y^{2}+4 x+2=0\) (C) \(x^{2}+4 y+2=0\) (D) \(x^{2}-4 y+2=0\)

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 (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

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) \(\left(-1, \frac{7}{3}\right)\) (B) \(\left(\frac{-1}{3}, \frac{7}{3}\right)\) (C) \(\left(1, \frac{7}{3}\right)\) (D) \(\left(\frac{1}{3}, \frac{7}{3}\right)\)

A straight line through the point \(A(3,4)\) is such that its intercept between the axes is bisected at \(A\). 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 locus of the vertices of the family of parabolas \(y=\frac{a^{3} x^{2}}{3}+\frac{a^{2} x}{2}-2 a\) is (A) \(x y=\frac{105}{64}\) (B) \(x y=\frac{3}{4}\) (C) \(x y=\frac{35}{16}\) (D) \(x y=\frac{64}{105}\)

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 \(a\) belongs to (A) \(\left(0, \frac{1}{2}\right)\) (B) \((3, \infty)\) (C) \(\left(\frac{1}{2}, 3\right)\) (D) \(\left(-3,-\frac{1}{2}\right)\)

Let \(A(h, k), B(1,1)\) and \(C(2,1)\) be the vertices of a right angled triangle with \(A C\) as its hypotenuse. If the area of the triangle is 1 , then the set of values which ' \(k\) ' can take is given by (A) \(\\{1,3\\}\) (B) \(\\{0,2\\}\) (C) \(\\{-1,3\\}\) (D) \(\\{-3,-2\\}\)

Let \(P=(-1,0), Q=(0,0)\) and \(R=(3,3 \sqrt{3})\) be three points. The equation of the bisector of the angle \(P Q R\) (A) \(\sqrt{3} x+y=0\) (B) \(x+\frac{\sqrt{3}}{2} y=0\) (C) \(\frac{\sqrt{3}}{2} x+y=0\) (D) \(x+\sqrt{3} y=0\)

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 \(x=0\) and \(y=0\), then \(m\) is (A) \(-\frac{1}{2}\) (B) \(-2\) (C) 1 (D) 2

The perpendicular bisector of the line segment joining \(P(1,4)\) and \(Q(k, 3)\) has \(y\)-intercept \(-4\). Then a possible value of \(k\) is (A) 1 (B) 2 (C) \(-2\) (D) \(-4\)

The lines \(L_{1}: y-x=0\) and \(L_{2}: 2 x+y=0\) intersect the line \(L_{3}: y+2=0\) at two respective points \(P\) and \(Q\). The bisector of the acute angle between \(L_{1}\) and \(L_{2}\) intersect \(L_{3}\) at \(R\). 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 \(-1\)

Equation of the ellipse which passes through the point \((-3,1)\), whose axes are the coordinate axes and has eccentricity \(\sqrt{\frac{2}{5}}\) is (A) \(5 x^{2}+3 y^{2}-48=0\) (B) \(3 x^{2}+5 y^{2}-15=0\) (C) \(5 x^{2}+3 y^{2}-32=0\) (D) \(3 x^{2}+5 y^{2}-32=0\)

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) \(\frac{11}{5}\)

A line is drawn through the point \((1,2)\) to meet the coordinate axes at points \(P\) and \(Q\) respectively such that it forms a triangle \(O P Q\), where \(O\) is the origin. If the area of the triangle \(O P Q\) is least, then the slope of the line \(P Q\) is (A) \(-\frac{1}{4}\) (B) \(-4\) (C) \(-2\) (D) \(-\frac{1}{2}\)

A ray of light along \(x+\sqrt{3 y}=\sqrt{3}\) gets reflected upon reaching \(x\)-axis, the equation of the reflected ray is (A) \(\sqrt{3 y}=x-\sqrt{3}\) (B) \(y=\sqrt{3 x}-\sqrt{3}\) (C) \(\sqrt{3 y}=x-1\) (D) \(y=x+\sqrt{3}\)

The abscissa 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 (A) \(2-\sqrt{2}\) (B) \(1+\sqrt{2}\) (C) \(1-\sqrt{2}\) (D) \(2+\sqrt{2}\)

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

Let \(P S\) be the median of the triangle with vertices \(P\) \((2,2), Q(6,-1)\) and \(R(7,3)\). The equation of the line passing through \((1,-1)\) and parallel to \(P S\) is (A) \(4 x-7 y-11=0\) (B) \(2 x+9 y+7=0\) (C) \(4 x+7 y+3=0\) (D) \(2 x-9 y-11=0\)

The number of points, having both co-ordinates as integers, which lie in the interior of the triangle with vertices \((0,0),(0,41)\) and \((41,0)\), is: (A) 861 (B) 820 (C) 780 (D) 901

Locus of the image of the point \((2,3)\) in the line \((2 x-\) \(3 y+4)+k(x-2 y+3)=0, k \in R\), is a: (A) straight line parallel to \(y\)-axis. (B) circle of radius 2 . (C) circle of radius 3 . (D) straight line parallel to \(x\)-axis.

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? (A) \(\left(-\frac{10}{3},-\frac{7}{3}\right)\) (B) \((-3,-9)\) (C) \((-3,-8)\) (D) \(\left(\frac{1}{3},-\frac{8}{3}\right)\)