Chapter 2

The line \(3 x+2 y=24\) meets \(y\) -axis at \(A\) and \(x\) -axis at \(B\). The perpendicular bisector of \(A B\) meets the line through \((0,-1)\) parallel to \(x\) -axis at \(C\). Find the area \(\Delta A B C\).

If \((x, y)\) be an arbitrary point on the altitude through \(A\) of \(\Delta A B C\) with vertices \(\left(x_{i}, y_{i}\right), i=1,2,3\) then the equation of the altitude through \(A\) is \(b \sec B\left|\begin{array}{lll}x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1\end{array}\right|+c \sec C\left|\begin{array}{lll}x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1\end{array}\right|=0\).

The straight line \(3 x+4 y=5\) and \(4 x-3 y=15\) intersect at the point \(A\). On this line, the points \(B\) and \(C\) are chosen so that \(A O=A C\). Find the possible equations of the line \(B C\) passing through the point \((1,2)\).

The consecutive sides of a parallelogram are \(4 x+5 y=0\) and \(7 x+2 y=0 .\) If the equation of one diagonal is \(11 x+7 y=9\), find the equation of the other diagonal.

Find the locus of the foot of the perpendicular from the origin upon the line joining the points \((a \cos \theta, b \sin \theta)\) and \((-a \sin \theta, b \cos \theta)\) where \(a\) is a variable.

Show that the locus given by \(x+y=0,(a-b) x+(a+b) y=2 a b\) and \((a+b) x+\) \((a-b) y=2 a b\) form an isosceles triangle whose vertical angle is \(2 \tan ^{-1}\left(\frac{a}{b}\right)\). Determine the centroid of a triangle.

Given \(n\) straight lines and a fixed point \(O\). Through \(O\) a straight line is drawn meeting these lines in the point \(A_{1}, A_{2}, \ldots, A_{n}\) and a point \(A\) such that \(\frac{n}{O A}=\frac{1}{O A_{1}}+\frac{1}{O A_{2}}+\cdots+\frac{1}{O A_{n}} .\) Prove that the locus of the point \(A\) is a straight line.

Find the equation of the line passing through the point \((2,3)\) and making intercepts of length 2 units and between the lines.

Prove that the points \((a, b),(c, d)\), and \((a-c, b-d)\) are collinear if \((a d=b c)\). Also, show that the straight line passing through these points passes through the origin.

One diagonal of a square is along the line \(8 x-15 y=0\) and one of its vertices is \((1,2)\). Find the equations of the sides of the square through this vertex.

The sides of a triangle are \(u_{r}=x \cos \alpha_{r}+y \sin \alpha-p_{r}=0, r=1,2,3\). Show that its orthocentre is given by \(u_{1} \cos \left(\alpha_{2}-\alpha_{3}\right)=u_{2} \cos \left(\alpha_{3}-\alpha_{1}\right)=u_{3} \cos \left(\alpha_{1}-\alpha_{2}\right)\).

Let a line \(L\) has intercepts \(a\) and \(b\) on the coordinate axes. When the axes are rotated through an angle, keeping the origin fixed, the same line \(L\) has intercepts \(p\) and \(q\). Obtain the relation between \(a, b, p\), and \(q\).

A line is such that its segment between the straight lines \(5 x-y-4=0\) and \(3 x+4 y-4=0\) is bisected at the point \((1,5)\). Obtain its equation.

Prove that the \((a-b) x+(b-c) y+(c-a)=0,(a-c) x+(c-a) y+(a-b)=0\), and \((c-a) x+(a-b) y+(b-c)=0\) are concurrent.

Two vertices of a triangle are \((5,-1)\) and \((-2,3)\). If the orthocentre of the triangle is at the origin, find the coordinates of the third vertex.

A line intersects \(x\) -axis at \(A(7,0)\) and \(y\) -axis at \(B(0,-5)\). A variable line \(P Q\) which is perpendicular to \(A B\) intersects \(x\) -axis at \(P\) and \(y\) -axis at \(Q .\) If \(A Q\) and BP intersect at \(R\), then find the locus of \(R\).

Show that the straight lines \(7 x-2 y+10=0,7 x+2 y-10=0\), and \(y=2\) form an isosceles triangle and find its area.

The equations of the sides \(B C, C A\), and \(A B\) of a triangle \(A B C\) are \(K_{r}=a_{r} x+\) \(b_{r} y+c_{r}=0, r=1,2,3\). Prove that the equation of a line drawn through \(A\) parallel to \(B C\) is \(K_{3}\left(a_{2} b_{1}-a_{1} b_{2}\right)=K_{2}\left(a_{3} b_{1}-a_{1} b_{3}\right)\).

The sides of a triangle \(A B C\) are determined by the equation \(u_{r}=a_{r} x+b_{r} y+\) \(c_{r}=0, r=1,2,3 .\) Show that the coordinates of the orthocentre of the triangle \(A B C\) satisfy the equation \(\lambda_{1} u_{1}=\lambda_{2} u_{2}+\lambda_{3} u_{3}\) where \(\lambda_{1}=a_{2} a_{3}+b_{2} b_{3}, \lambda_{2}=a_{3} a_{1}+\) \(b_{3} b_{1}\), and \(\lambda_{3}=a_{1} a_{2}+b_{1} b_{2} .\)

Prove that the two lines can be drawn through the point \(P(P, Q)\) so that their perpendicular distances from the point \(Q(2 a, 2 a)\) will be equal to \(a\) and find their equations.

Find the locus of a point which moves such that the square of its distance from the base of an isosceles triangle is equal to the rectangle under its distances from the other sides.

Prove that the lines given by \((b+c) x-b c y=a\left(b^{2}+b c+c^{2}\right),(c+a) x-c a y=\) \(b\left(c^{2}+c a+a^{2}\right)\), and \((a+b) x-a b y=c\left(a^{2}+a b+b^{2}\right)\) are concurrent.

Show that the area of the triangle formed by the lines \(y=m_{1} x+c_{1}, y=m_{2} x+\) \(c_{2}\), and \(y=m_{3} x+c_{3}\) is \(\frac{1}{2}\left[\frac{\left(c_{2}-c_{3}\right)^{2}}{m_{2}-m_{3}}+\frac{\left(c_{3}-c_{1}\right)^{3}}{m_{3}-m_{1}}+\frac{\left(c_{1}-c_{2}\right)^{2}}{m_{1}-m_{2}}\right] .\)

Find the bisector of the acute angle between the lines \(3 x+4 y=1\) which is the bisector containing the origin.

Find the equation to the diagonals of the parallelogram formed by the lines \(a x+b y+c=0, a x+b y+d=0, a^{\prime} x+b^{\prime} y+c^{\prime}=0, a^{\prime} x+b^{\prime} y-d^{\prime}=0 .\) Show that the parallelogram will be a rhombus if \(\left(a^{2}+b^{2}\right)\left(c^{\prime}-d^{\prime}\right)^{2}=\left(a^{\prime 2}+b^{\prime 2}\right)(c-d)^{2}\)

A variable line is at a constant distance \(p\) from the origin and meets coordinate axes in \(A\) and \(B\). Show that the locus of the centroid of the \(\Delta O A B\) is \(x^{-2}+y^{-2}=p^{-2} .\)

A moving line is \(l x+m y+n=0\) where \(l, m\), and \(n\) are connected by the relation \(a l+b m+c n=0\), and \(a, b\), and \(c\) are constants. Show that the line passes through a fixed point.

Find the equation of bisector of acute angle between the lines \(3 x-4 y+7=0\) and \(12 x+5 y-2=0\).

The lines \(a x+b y+c=0, b x+c y+a=0\), and \(c x+a y+b=0\) are concurrent where \(a, b\), and \(c\) are the sides of the \(\Delta A B C\) in usual notation and prove that \(\sin ^{3} A+\sin ^{3} B+\sin ^{3} C=3 \sin A \sin B \sin C .\)

Let \(\triangle A B C\) be a triangle with \(A B=A C .\) If \(D\) is the midpoint of \(B C\), and \(E\) is the foot of the perpendicular drawn from \(D\) to \(A C\) and \(F\) is the midpoint of \(B E\). Prove that \(A F\) is perpendicular to \(B E\).

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

Prove that the diagonals of the parallelogram formed by the lines \(a x+b y+c=0, a x+b y+c^{\prime}=0, a^{\prime} x+b^{\prime} y+c=0\), and \(a^{\prime} x+b^{\prime} y+c^{\prime}=0\) will be at right angles if \(a^{2}+b^{2}=a^{\prime 2}+b^{\prime 2}\).

One diagonal of a square is the portion of the line \(\frac{x}{a}+\frac{y}{b}=1\) intercepted between the axes. Show that the extremities of the other diagonal are \(\left(\frac{a+b}{2}, \frac{a+b}{2}\right)\) and \(\left(\frac{a-b}{2}, \frac{a-b}{2}\right)\).

Show that the origin lies inside a triangle whose vertices are given by the equations \(7 x-5 y-11=0,8 x+3 y+31=0\), and \(x+3 y-19=0\).

If the lines \(p_{1} x+q_{1} y=1, p_{2} x+q_{2} y=1\), and \(p_{3} x+q_{3} y=1\) are concurrent, prove that the points \(\left(p_{1}, q_{1}\right),\left(p_{2}, q_{2}\right)\), and \(\left(p_{3}, q_{3}\right)\) are collinear.

If \(p, q\), and \(r\) be the length of the perpendiculars from the vertices \(A, B\), and \(C\) of a triangle on any straight line, prove that \(a^{2}(p-q)(p-r)+b^{2}(q-r)(q-p)+\) \(c^{2}(r-p)(r-q)=4 \Delta^{2} .\)

Prove that the area of the parallelogram formed by the straight line \(a_{1} x+b_{1} y+\) \(c_{1}=0, a_{1} x+b_{1} y+d_{1}=0, a_{2} x+b_{2} y+c_{2}=0\), and \(a_{2} x+b_{2} y+d_{2}=0\) is \(\left|\frac{\left(d_{1}-c_{1}\right)\left(d_{2}-c_{2}\right)}{a_{1} b_{2}-a_{2} b_{1}}\right|\).

Two sides of an isosceles triangle are given by the equations \(7 x-y+3=0\) and \(x+y-7=0\) and its third side passes through the point \((1,-10)\). Determine the equation of the third side.

Are the points \((3,4)\) and \((2,-6)\) on the same or opposite sides of the line \(3 x-4 y=8 ?\)

How many circles can be drawn each touching all the three lines \(x+y=1, y=x\), and \(7 x-y=6 ?\) Find the centre and radius of one of the circles.

Show that \(P\left(1+\frac{t}{\sqrt{2}}, 2+\frac{t}{\sqrt{2}}\right)\) be any point on a line then the range of values of \(t\) for which the point \(p\) lies between the parallel lines \(x+2 y=1\) and \(2 x+\) \(4 y=15\) is \(\left(\frac{-4 \sqrt{2}}{5}, \frac{5 \sqrt{2}}{6}\right)\).