Chapter 18

If one of the diagonals of a square is along the line \(x=\) \(y\) and one of its vertices is \((3,0)\), then its side through this vertex nearer to the origin is given by the equation. (A) \(y-3 x+9=0\) (B) \(3 y+x-3=0\) (C) \(x-3 y-3=0\) (D) \(3 x+y-9=0\)

Through the point \(P(\alpha, \beta)\), where \(a \beta>0\) the straight line \(\frac{x}{a}+\frac{y}{b}=1\) is drawn so as to form with coordinate axes a triangle of area \(S\). If \(a b>0\), then the least value of \(S\) is (A) \(\alpha \beta\) (B) \(2 \alpha \beta\) (C) \(4 \alpha \beta\) (D) none of these

A line joining two points \(A(2,0)\) and \(B(3,1)\) is rotated about \(A\) in anti- clockwise direction through an angle \(15^{\circ} .\) If \(B\) goes to \(C\) in the new position, then the coordinates of \(C\) are (A) \(\left(2, \sqrt{\frac{3}{2}}\right)\) (B) \(\left(2,-\sqrt{\frac{3}{2}}\right)\) (C) \(\left(2+\frac{1}{\sqrt{2}}, \sqrt{\frac{3}{2}}\right)\) (D) none of these

\(P\) is a point on either of the two lines \(y-\sqrt{3}|x|=2\) at a distance of 5 units from their point of intersection. The coordinates of the foot of the perpendicular from \(P\) on the bisector of the angle between them are (A) \(\left[0, \frac{1}{2}(4+5 \sqrt{3})\right]\) or \(\left[0, \frac{1}{2}(4-5 \sqrt{3})\right]\) depend- ing on which line the point \(P\) is taken (B) \(\left[0, \frac{1}{2}(4+5 \sqrt{3})\right]\) (C) \(\left[0, \frac{1}{2}(4-5 \sqrt{3})\right]\) (D) \(\left[\frac{5}{2}, \frac{5 \sqrt{3}}{2}\right]\)

The condition to be imposed on \(\beta\) so that \((0, \beta)\) lies on or inside the triangle having sides \(y+3 x+2=0\), \(3 y-2 x-5=0\) and \(4 y+x-14=0\) is (A) \(0<\beta<\frac{5}{3}\) (B) \(0<\beta<\frac{7}{2}\) (C) \(\frac{5}{3} \leq \beta \leq \frac{7}{2}\) (D) none of these

The point \((1, \beta)\) lies on or inside the triangle formed by the lines \(y=x, x\)-axis and \(x+y=8\), if (A) \(0<\beta<1\) (B) \(0 \leq \beta \leq 1\) (C) \(0<\beta<8\) (D) none of these

A ray of light travelling along the line \(x+\sqrt{3} y=5\) is incident on the \(x\)-axis and after refraction it enters the other side of the \(x\)-axis by turning \(\frac{\pi}{6}\) away from the \(x\)-axis. The equation of the line along which the refracted ray travels is (A) \(x+\sqrt{3} y-5 \sqrt{3}=0\) (B) \(x-\sqrt{3} y-5 \sqrt{3}=0\) (C) \(\sqrt{3} x+y-5 \sqrt{3}=0\) (D) \(\sqrt{3} x-y-5 \sqrt{3}=0\)

A ray of light is sent along the line which passes through the point \((2,3)\). The ray is reflected from the point \(P\) on \(x\)-axis. If the reflected ray passes through the point \((6,4)\), then the coordinates of \(P\) are (A) \(\left(\frac{26}{7}, 0\right)\) (B) \(\left(0, \frac{26}{7}\right)\) (C) \(\left(-\frac{26}{7}, 0\right)\) (D) none of these

A line passing through the point \(P(4,2)\), meets the \(x\)-axis and \(y\)-axis at \(A\) and \(B\), respectively. If \(O\) is the origin, then locus of the centre of the circum circle of \(\Delta O A B\) is (A) \(x^{-1}+y^{-1}=2\) (B) \(2 x^{-1}+y^{-1}=1\) (C) \(x^{-1}+2 y^{-1}=1\) (D) \(2 x^{-1}+2 y^{-1}=1\)

If the point \((2 \cos \theta, 2 \sin \theta)\) does not fall in that angle between the lines \(y=|x-2|\) in which the origin lies then \(\theta\) belongs to (A) \(\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right)\) (B) \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) (C) \((0, \pi)\) (D) none of these

A line is drawn from the point \(P(\alpha, \beta)\), making an angle \(\theta\) with the positive direction of \(x\)-axis, to meet the line \(a x+b y+c=0\) at \(Q .\) The length of \(P Q\) is (A) \(-\frac{a \alpha+b \beta+c}{a \cos \theta+b \sin \theta}\) (B) \(\left|\frac{a \alpha+b \beta+c}{\sqrt{a^{2}+b^{2}}}\right|\) (C) \(\frac{a \alpha+b \beta+c}{a \cos \theta+b \sin \theta}\) (D) none of these

If the equal sides \(A B\) and \(A C\) (each equal to \(a\) ) of a right-angled isosceles triangle \(A B C\) be produced to \(P\) and \(Q\) so that \(B P \cdot C Q=A B^{2}\), then the line \(P Q\) always passes through the fixed point (A) \((a, 0)\) (B) \((0, a)\) (C) \((a, a)\) (D) none of these

If \(x_{1}, x_{2}, x_{3}\) as well as \(y_{1}, y_{2}, y_{3}\) are 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) lie on a straight line (B) lie on an ellipse (C) lie on a circle (D) are vertices of a triangle

If the distance of any point \(P(x, y)\) from the origin is defined as \(d(x, y)=\operatorname{Max} .\\{|x|,|y|\\}\) and \(d(x, y)=k\) (nonzero constant), then the locus of the point \(P\) is (A) a straight line (B) a circle (C) a parabola (D) none of these

If \(a, b, c\) form an A. P. with common difference \(d(\neq 0)\) and \(x, y, z\) form a G. P. with common ratio \(r(\neq 1)\), then the area of the triangle with vertices \((a, x),(b, y)\) and \((c, z)\) is independent of (A) \(b\) (B) \(r\) (C) \(d\) (D) \(x\)

A line of fixed length 2 units moves so that its ends are on the positive \(x\)-axis and that part of the line \(x+y=\) 0 which lies in the second quadrant. The locus of the mid-point of the line has the equation (A) \((x+2 y)^{2}+y^{2}=1\) (B) \((x-2 y)^{2}+y^{2}=1\) (C) \((x+2 y)^{2}-y^{2}=1\) (D) none of these

A straight line through the origin \(O\) meets the parallel lines \(4 x+2 y=9\) and \(2 x+y+6=0\) at points \(P\) and \(Q\), respectively. The point \(O\) divides the segment \(P Q\) in the ratio (A) \(1: 2\) (B) \(3: 4\) (C) \(2: 1\) (D) \(4: 3\)

Let \(O\) be the origin and let \(A(2,0), B(0,2)\) be two points. If \(P(x, y)\) is a point such that \(x y>0\) and \(x+y<\) 2 , then (A) \(P\) lies either inside the triangle \(O A B\) or in the third quadrant (B) \(P\) cannot be inside the triangle \(O A B\) (C) \(P\) lies inside the triangle \(O A B\) (D) none of these

Consider the equation \(y-y_{1}=m\left(x-x_{1}\right) .\) In this equation, if \(m\) and \(x_{1}\) are fixed and different lines are drawn for different values of \(y^{1}\), then, (A) the lines will pass through a single point (B) there will be one possible line only (C) there will be a set of parallel lines (D) none of these

\(D\) is a point on \(A C\) of the triangle with vertices \(A(2,\), 3), \(B(1,-3), C(-4,-7)\) and \(B D\) divides \(A B C\) into two triangles of equal area. The equation of the line drawn through \(B\) at right angles to \(B D\) is (A) \(y-2 x+5=0\) (B) \(2 y-x+5=0\) (C) \(y+2 x-5=0\) (D) \(2 y+x-5=0\)

If two points \(A(a, 0)\) and \(B(-a, 0)\) are stationary and if \(\angle A-\angle B=\theta\) in \(\triangle A B C\), the locus of \(C\) is (A) \(x^{2}+y^{2}+2 x y \tan \theta=a^{2}\) (B) \(x^{2}-y^{2}+2 x y \tan \theta=a^{2}\) (C) \(x^{2}+y^{2}+2 x y \cot \theta=a^{2}\) (D) \(x^{2}-y^{2}+2 x y \cot \theta=a^{2}\)

If the point \(P\left(a^{2}, a\right.\) ) lies in the region corresponding to the acute angle between the lines \(2 y=x\) and \(4 y=x\), then (A) \(a \in(2,6)\) (B) \(a \in(4,6)\) (C) \(a \in(2,4)\) (D) none of these

The point \((4,1)\) undergoes the following three successive transformations (A) Reflection about the line \(y=x-1\) (B) Translation through a distance 1 unit along the positive \(x\)-axis (C) Rotation through an angle \(\frac{\pi}{4}\) about the origin in the anti- clockwise direction. Then, the coordinates of the final point are (A) \((4,3)\) (B) \(\left(\frac{7}{2}, \frac{7}{2}\right)\) (C) \((0,3 \sqrt{2})\) (D) \((3,4)\)

The distance between two parallel lines is unity. A point \(P\) lies between the lines at a distance \(a\) from one of them. The length of a side of an equilateral triangle \(P Q R\), vertex \(Q\) of which lies on one of the parallel lines and vertex \(R\) lies on the other line, is (A) \(\frac{2}{\sqrt{3}} \cdot \sqrt{a^{2}+a+1}\) (B) \(\frac{2}{\sqrt{3}} \sqrt{a^{2}-a+1}\) (C) \(\frac{1}{\sqrt{3}} \sqrt{a^{2}+a+1}\) (D) \(\frac{1}{\sqrt{3}} \sqrt{a^{2}-a+1}\)

Two points \(A\) and \(B\) are given. \(P\) is a moving point on one side of the line \(A B\) such that \(\angle P A B-\angle P B A\) is a positive constant \(2 \theta\). The locus of the point \(P\) is (A) \(x^{2}+y^{2}+2 x y \cot 2 \theta=a^{2}\) (B) \(x^{2}+y^{2}-2 x y \cot 2 \theta=a^{2}\) (C) \(x^{2}+y^{2}+2 x y \tan 2 \theta=a^{2}\) (D) \(x^{2}-y^{2}+2 x y \cot 2 \theta=a^{2}\).

The four points \(A(p, 0), B(q, 0), C(r, 0)\) and \(D(s, 0)\) are such that \(p, q\) are the roots of the equation \(a x^{2}+2 h x+\) \(b=0\) and \(r, s\) are those of equation \(a^{\prime} x^{2}+2 h^{\prime} x+b^{\prime}=0\). If the sum of the ratios in which \(C\) and \(D\) divide \(A B\) is zero, then (A) \(a b^{\prime}+a^{\prime} b=2 h h^{\prime}\) (B) \(a b^{\prime}+a^{\prime} b=h h^{\prime}\) (C) \(a b^{\prime}-a^{\prime} b=2 h h^{\prime}\) (D) none of these

The coordinates of a point \(P\) on the line \(3 x+2 y+10\) \(=0\) such that \(|P A-P B|\) is maximum where \(A\) is \((4,2)\) and \(B\) is \((2,4)\), are (A) \((22,28)\) (B) \((22,-28)\) (C) \((-22,28)\) (D) \((-22,-28)\)

A line through \(A(-5,-4)\) meets the lines \(x+3 y+2=0\), \(2 x+y+4=0\) and \(x-y-5=0\) at the point \(B, C\) and \(D\), respectively. If \(\left(\frac{15}{A B}\right)^{2}+\left(\frac{10}{A C}\right)^{2}=\left(\frac{6}{A D}\right)^{2}\), the equa- tion of the line is (A) \(2 x+3 y+22=0\) (B) \(2 x-3 y+22=0\) (C) \(3 x+2 y+22=0\) (D) \(3 x-2 y+22=0\)

\(A(0,0), B(2,1)\) and \(C(3,0)\) are the vertices of a \(\triangle A B C\) and \(B D\) is its altitude. If the line through \(D\) parallel to the side \(A B\) intersects the side \(B C\) at a point \(K\), then the product of the areas of the triangles \(A B C\) and \(B D K\) is (A) 1 (B) \(\frac{1}{2}\) (C) \(\frac{1}{4}\) (D) none of these

A line cuts the \(x\)-axis at \(A(7,0)\) and \(y\)-axis at \(B(0,-5)\). A variable line \(P Q\) is drawn \(\perp\) to \(A B\) cutting the \(x\)-axis in \(P\) and the \(y\)-axis in \(Q .\) If \(A Q\) and \(B P\) intersect at \(R\), then the locus of \(R\) is (A) \(x(x-7)+y(y+5)=0\) (B) \(x(x-7)-y(y+5)=0\) (C) \(x(x+7)+y(y-5)=0\) (D) none of these

The point \((2,3)\) undergoes the following three transformations successively (i) reflection about the line \(y=x\) (ii) translation through a distance 2 units along the positive direction of \(y\)-axis (iii) rotation through an angle of \(45^{\circ}\) about the origin in the anti- clockwise direction. The final coordinates of the point are (A) \(\left(\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}}\right)\) (B) \(\left(-\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}}\right)\) (C) \(\left(\frac{1}{\sqrt{2}},-\frac{7}{\sqrt{2}}\right)\) (D) none of these

The equations of the perpendicular bisector of the sides \(A B\) and \(A C\) of a \(\Delta A B C\) are \(x-y+5=0\) and \(x+\) \(2 y=0\), respectively. If the point \(A\) is \((1,-2)\) then the equation of the line \(B C\) is (A) \(14 x+23 y=40\) (B) \(14 x-23 y=40\) (C) \(23 x+14 y=40\) (D) \(23 x-14 y=40\)

The equation of a family of lines is given by \((2+3 t)\) \(x+(1-2 t) y+4=0\), where \(t\) is the parameter. The equation of a straight line, belonging to this family, at the maximum distance from the point \((2,3)\) is (A) \(21 x+14 y=0\) (B) \(21 x-14 y=0\) (C) \(14 x-21 y=0\) (D) none of these

\(A B C D\) is a square whose vertices \(A, B, C\) and \(D\) are \((0,0),(2,0),(2,2)\) and \((0,2)\), respectively. This square is rotated in the \(X-Y\) plane with an angle of \(30^{\circ}\) in anti-clockwise direction about an axis passing through the vertex \(A\). The equation of the diagonal \(B D\) of this rotated square is (A) \(\sqrt{3} x+(1-\sqrt{3}) y=\sqrt{3}\) (B) \((1+\sqrt{3}) x-(1-\sqrt{2})=2\) (C) \((2-\sqrt{3}) x+y=2(\sqrt{3}-1)\) (D) none of these

The equations of the straight lines passing through \((-2,-7)\) and cutting an intercept of length three units between the straight lines \(4 x+3 y=12\) and \(4 x+3 y=\) 3 are (A) \(x+2=0, y+7=\frac{7}{24}(x+2)\) (B) \(x-2=0, y+7=-\frac{7}{24}(x+2)\) (C) \(x+2=0, y+7=-\frac{7}{24}(x+2)\) (D) \(x+2=0, y+7=-\frac{7}{12}(x+2)\)

The coordinates of the point which is at unit distance from the lines \(L_{1} \equiv 3 x-4 y+1=0\) and \(L_{2} \equiv 8 x+6 y+\) \(1=0\) and lies below \(L_{1}\) and above \(L_{2}\) are (A) \(\left(\frac{6}{5}, \frac{1}{10}\right)\) (B) \(\left(\frac{6}{5},-\frac{1}{10}\right)\) (C) \(\left(\frac{6}{5}, \frac{1}{5}\right)\) (D) \(\left(\frac{6}{5},-\frac{1}{5}\right)\)

The vertices of a triangle are \(A\left(x_{1}, x_{1} \tan \alpha\right), B\left(x_{2}, x_{2}\right.\) \(\tan \beta\) ) and \(C\left(x_{3}, x_{3} \tan \gamma\right)\). If the circumcentre of triangle \(A B C\) coincides with the origin and \(H(a, b)\) be its orthocentre then \(\frac{a}{h}=\) (A) \(\frac{\cos \alpha+\cos \beta+\cos \gamma}{\cos \alpha \cdot \cos \beta \cdot \cos \gamma}\) (B) \(\frac{\sin \alpha+\sin \beta+\sin \gamma}{\sin \alpha \cdot \sin \beta \cdot \sin \gamma}\) (C) \(\frac{\tan \alpha+\tan \beta+\tan \gamma}{\tan \alpha \cdot \tan \beta \cdot \tan \gamma}\) (D) \(\frac{\cos \alpha+\cos \beta+\cos \gamma}{\sin \alpha+\sin \beta+\sin \gamma}\)

\(O X\) and \(O Y\) are two coordinate axes. On \(O Y\) is taken a fixed point \(P\) and on \(O X\) any point \(Q .\) On \(P Q\) an equilateral triangle is described, its vertex \(R\) being on the side of \(P Q\) away from \(O\), then the locus of \(R\) will be (A) straight line (B) circle (C) ellipse (D) parabola

If the vertices of a variable triangle are \((3,4),(5 \mathrm{cos}\) \(\theta, 5 \sin \theta\) ) and \((5 \sin \theta,-5 \cos \theta)\), then the locus of its orthocentre is (A) \((x+y-1)^{2}+(x-y-7)^{2}=100\) (B) \((x+y-7)^{2}+(x-y+1)^{2}=100\) (C) \((x+y-7)^{2}+(x-y-1)^{2}=100\) (D) \((x+y+7)^{2}+(x+y-1)^{2}=100\)

The line \(x+y=1\) meets \(x\)-axis at \(A\) and \(y\)-axis at \(B \cdot P\) is the mid- point of \(A B \cdot P_{1}\) is the foot of the perpendicular from \(P\) to \(O A ; M_{1}\) is that from \(P_{1}\) to \(O P ; P_{2}\) is that from \(M_{1}\) to \(O A\) and so on. If \(P_{n}\) denotes the foot of the \(n\)th perpendicular on \(O A\) from \(M_{n-1}\), then \(O P_{n}\) is equal to (A) \(\frac{1}{2^{n}}\) (B) \(\frac{1}{2^{n-1}}\) (C) \(\frac{1}{2^{n-2}}\) (D) none of these

The line \(x+y=a\) meets \(x\)-axis at \(A\). A triangle \(A M N\) is inscribed in the triangle \(O A B, O\) being the origin with right angle at \(N ; M\) and \(N\) lie respectively on \(O B\) and \(A B\). If area of \(\Delta A M N\) is \(\frac{3}{8}\) of the area of triangle \(O A B\), then \(\frac{A N}{B N}\) is equal to (A) 3 (B) \(\frac{1}{3}\) (C) 2 (D) \(\frac{2}{3}\)

Let \(S_{1}, S_{2}, \ldots\) be squares such that for each \(n \geq 1\), the length of a side of \(S_{n}\) equals the length of a diagonal of \(S_{n+1} .\) If the length of a side of \(S_{1}\) is \(10 \mathrm{~cm}\), then for which of the following values of \(n\) is the area of \(S_{n}\) less than 1 square \(\mathrm{cm} ?\) (A) 7 (B) 8 (C) 9 (D) 10

A line which makes an acute angle \(\theta\) with the positive direction of \(x\)-axis is drawn through the point \(P(3,4)\) to meet the line \(x=6\) at \(R\) and \(y=8\) at \(S\), then (A) \(P R=3 \sec \theta\) (B) \(P S=4 \operatorname{cosec} \theta\) (C) \(P R+P S=\frac{2(3 \sin \theta+4 \cos \theta)}{\sin 2 \theta}\) (D) \(\frac{9}{(P R)^{2}}+\frac{16}{(P S)^{2}}=1\)

Straight lines \(3 x+4 y=5\) and \(4 x-3 y=15\) intersect at A. Points \(B\) and \(C\) are choosen on these lines such that \(A B=A C .\) The equation of the line \(B C\) passing through the point \((1,2)\) is (A) \(x+7 y+13=0\) (B) \(x-7 y+13=0\) (C) \(7 x+y-9=0\) (D) none of these

The equation of the straight line passing through the point \((4,5)\) and making equal angles with the two straight lines given by the equations \(3 x-4 y-7=0\) and \(12 x-5 y+6=0\), is (A) \(9 x-7 y-1=0\) (B) \(9 x+7 y-1=0\) (C) \(7 x+9 y-73=0\) (D) \(7 x+9 y+73=0\)

Let the algebraic sum of the perpendicular distances from the points \(A(2,0), B(0,2), C(1,1)\) to a variable line be zero. Then, all such lines (A) are concurrent (B) pass through the fixed point \((1,1)\) (C) touch some fixed circle (D) pass through the centroid of \(\triangle A B C\)

The equation of the line passing through the point (2, 3) and making intercept of length 2 units between the lines \(y+2 x=3\) and \(y+2 x=5\), is (A) \(x=2\) (B) \(3 x+4 y=18\) (C) \(4 x+3 y=18\) (D) none of these

Two sides of a rhombus \(A B C D\) are parallel to the lines \(y=x+2\) and \(y=7 x+3\). If the diagonals of the rhombus intersect at the point \((1,2)\) and the vertex \(A\) is on the \(y\)-axis, then the possible coordinates of \(A\) are (A) \((0,0)\) (B) \(\left(0, \frac{5}{2}\right)\) (C) \(\left(0,-\frac{5}{2}\right)\) (D) none of these

The equations of two equal sides \(A B\) and \(A C\) of an isosceles triangle \(A B C\) are \(x+y=5\) and \(7 x-y=3\), respectively. The equation of the side \(B C\), if the area of \(\triangle A B C\) is 5 units, is (A) \(3 x+y-2=0\) (B) \(3 x+y-12=0\) (C) \(x-3 y+1=0\) (D) \(x-3 y+21=0\)

If the equation of the mirror be \(2 x+y-6=0\) and a ray passing through \((3,10)\) after being reflected by the mirror passes through \((7,2)\), then the equations of the incident ray and the reflected ray are (A) \(x+3 y-13=0\) (B) \(3 x-y+1=0\) (C) \(x-3 y+13=0\) (D) \(3 x+y-1=0\)