Let `l_1a n dl_2` be the two skew lines. If `P ,Q` are two distinct points on `l_1n dR , S` are two distinct points on `l_2,` then prove that `P R` cannot be parallel to `Q Sdot`

If a point Q lies between two points P and R such that PQ = QR, prove that PQ = 1/2 PR.

Let l_(1) and l_(2) be thw two lines intersecting at P. If A_(1), B_(1), C_(1) are points on l_(1), A_(2), B_(2), C_(2), D_(2), E_(2) are points on l_(2) an dif none of these coincides with P, then the number of triangles formed by joining these eight points is-

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

If P is any point on the plane l x+m y+n z=pa n dQ is a point on the line O P such that O P.O Q=p^2 , then find the locus of the point Qdot

Two circles intersect each other at the points G and H . A straight line is drawn through the point G which intersect two circles at the points P and Q and the straight line through the point H parallel to PQ intersects the two circles at the points R and S . Prove that PQ=RS .

If p be the first of n arithmetic means between two numbers and q be the first of n harmonic means between the same two numbers , then prove that the value of q can not lie between p and ((n+1)/(n-1))^2p .

Let vec(a) " and " vec(b) be the position vectors of two given points P and Q respectively. To find the position vector of the point R which divides the line segment bar(PQ) internally in the ratio m:n.

Two chords AC and BD of a circle intersect each other at the point O. If two tangents drawn at the points A and B intersect each other at the point P and two tangents drawn at the points C and D intersect at the point Q, prove that angle P+ angle Q =2 angle BOC. [GP-X]

A line L passing through the focus of the parabola y^2=4(x-1) intersects the parabola at two distinct points. If m is the slope of the line L , then