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 it always passes through a fixed point, O being the point of intersection of the lines

A straight line cuts intercepts from the axes of coordinates the sum of whose reciprocals is a constant. Show that it always passes though as fixed point.

A straight line moves so that the sum of the reciprocals of its intercepts made on axes is constant. Show that the line passes through a 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

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

Assertion: Number of triangles formed by n concurrent lines and a line which is parallel to one of the n concurrent lines is "^(n-1)C_2. Reason: One and only one triangle is formed by three lines out of which 2 are concurrent and third line is not passing through the point of intersection of the concurrent lines and not parallel to any of the concurrent lines. (A) Both A and R are true and R is the correct explanation of A (B) Both A and R are true R is not the correct explanation of A (C) A is true but R is false. (D) A is false but R is true.

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

Let a x+b y+c=0 be a variable straight line, whre a , ba n dc are the 1st, 3rd, and 7th terms of an increasing AP, respectively. Then prove that the variable straight line always passes through a fixed point. Find that point.

Let a x+b y+c=0 be a variable straight line, whre a , ba n dc are the 1st, 3rd, and 7th terms of an increasing AP, respectively. Then prove that the variable straight line always passes through a fixed point. Find that point.

Two straight lines pass through the fixed points (+-a, 0) and have slopes whose products is pgt0 Show that the locus of the points of intersection of the lines is a hyperbola.

Two straight lines pass through the fixed points (+-a, 0) and have slopes whose products is pgt0 Show that the locus of the points of intersection of the lines is a hyperbola.