Find x such that PQ=QR where `P(6,-1)Q(1,3) and R(x,8)` respectively.

Find the unit vector in the direction of vector vec(PQ) , where P and Q are the points (1, 2, 3) and (4, 5, 6) respectively.

The vertices of trianglePQR are P(0,-4),Q(3,1) and R(-8,1) Find the coordinates of N, the mid-point of QR.

The lengths of the tangents from P(1,-1) and Q(3,3) to a circle are sqrt(2) and sqrt(6) , respectively. Then, find the length of the tangent from R(-1,-5) to the same circle.

PQRS is a rectangle formed by joining the points P(-1, -1), Q(-1, 4), R(5,4) and S(5, -1) . A, B, C and D are the mid points of PQ, QR, RS and SR respectively. Is the quadrilateral ABCD a square, a rectangle or a rhombus? Justify your answer.

Let C_1 and C_2 be parabolas x^2 = y - 1 and y^2 = x-1 respectively. Let P be any point on C_1 and Q be any point C_2 . Let P_1 and Q_1 be the reflection of P and Q, respectively w.r.t the line y = x then prove that P_1 lies on C_2 and Q_1 lies on C_1 and PQ >= [P P_1, Q Q_1] . Hence or otherwise , determine points P_0 and Q_0 on the parabolas C_1 and C_2 respectively such that P_0 Q_0 <= PQ for all pairs of points (P,Q) with P on C_1 and Q on C_2

The vertices of trianglePQR are P(0,-4),Q(3,1) and R(-8,1) Find the area of trianglePQR .

The vertices of trianglePQR are P(0,-4),Q(3,1) and R(-8,1) Find the area of triangleMPN .

A rectangle PQRS has its side PQ parallel to the line y= mx and vertices P, Q , and S on the lines y= a, x=b ,and x= - b , respectively. Find the locus of the vertex R .