A variable line cuts n given concurrent straight lines at `A_1,A_2...A_n` such that `sum_(i=1)^n 1/(OA_i)` is a constant. Show that A,A , A such it always passes through a fixed point, O being the point of intersection of the lines

If A, B and C are in AP, then the straight line Ax + 2By+C =0 will always pass through a fixed point. The fixed point is

A st.line cuts intercepts from the axes,the sum of the reciprocals of which equals 4. Show that it always passes through a fixed pt.Find that fixed point.

Let L_1=0a n dL_2=0 be two fixed lines. A variable line is drawn through the origin to cut the two lines at R and SdotPdot is a point on the line A B such that ((m+n))/(O P)=m/(O R)+n/(O S)dot Show that the locus of P is a straight line passing through the point of intersection of the given lines R , S , R are on the same side of O)dot

There are n straight lines in a plane, no two of which are parallel and no three of which pass through the same point. How many additional lines can be generated by means of point of intersections of the given lines.

A circle passes through a fixed point A and cuts two perpendicular straight lines through A in B and C. If the straight line BC passes through a fixed-point (x_(1), y_(1)) , the locus of the centre of the circle, is

A variable line is drawn through O to cut two fixed straight lines L_(1) and L_(2) in R and S.A point P is chosen the variable line such (m+n)/(OP)=(m)/(OR)+(n)/(OS) Find the locus of P which is a straight ine passing through the point of intersection of L_(1) and L_(2)

Two straight lines pass through the fixed points (+-a,0) and have slopes whose products is p>0 show that the locus of the points of intersection of the lines is a hyperbola.

If the algebraic sum of perpendiculars from n given points on a variable straight line is zero then prove that the variable straight line passes through a fixed point