In a triangle `P Q R ,Sa n dT` are points on `Q Ra n dP R ,` respectively, such that `Q S=3S Ra n dP T=4T Rdot` Let `M` be the point of intersection of `P Sa n dQ Tdot` Determine the ratio `Q M : M T` using the vector method .

In a triangle A B C ,Da n dE are points on B Ca n dA C , respectivley, such that B D=2D Ca n dA E=3E Cdot Let P be the point of intersection of A Da n dB Edot Find B P//P E using the vector method.

In triangle A B C , a point P is taken on A B such that A P//B P=1//3 and point Q is taken on B C such that C Q//B Q=3//1 . If R is the point of intersection of the lines A Qa n dC P , using vector method, find the area of A B C if the area of B R C is 1 unit

The distance of two points P and Q on the parabola y^(2) = 4ax from the focus S are 3 and 12 respectively. The distance of the point of intersection of the tangents at P and Q from the focus S is

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

Three points P,Q ,R lie on a circle. The two perpendiculars PQ and PR and the point P intersect the circle at the points S and T respectively. Prove that RQ=ST

A B C is a triangle, P is a point on A B a n dQ is a point on A C such that /_A Q P=/_A B Cdot Complete the relation (A r e aof A P Q)/(A r e aof A B C)=dot(())/(A C^2) .

O P Q R is a square and M ,N are the middle points of the sides P Qa n dQ R , respectively. Then the ratio of the area of the square to that of triangle O M N is 4:1 (b) 2:1 (c) 8:3 (d) 7:3

If the vertices Pa n dQ of a triangle P Q R are given by (2, 5) and (4,-11) , respectively, and the point R moves along the line N given by 9x+7y+4=0 , then the locus of the centroid of triangle P Q R is a straight line parallel to PQ (b) QR (c) RP (d) N

The line x+y=p meets the x- and y-axes at Aa n dB , respectively. A triangle A P Q is inscribed in triangle O A B ,O being the origin, with right angle at QdotP and Q lie, respectively, on O Ba n dA B . If the area of triangle A P Q is 3/8t h of the are of triangle O A B , the (A Q)/(B Q) is equal to (a) 2 (b) 2/3 (c) 1/3 (d) 3

The points P,Q,R are collinear , if the coordinates of P and Q be (3,4) and (7,7) respectively and overline(PR)=10 unit , find the coordinates of R.