If the line segment joining the points P(x₁, y₁) and Q(x₂, y₂) subtends an angle α at the origin O, prove that : OP. OQ cos α = x₁ x₂ + y₁ y₂.

The vertices of a triangle ABC are A(0, 0), B (2, -1) and C (9, 0). Find cos B.

Four points A (6, 3), B(-3, 5), C(4, -2) and D(x, 3x) are given in such a way that , find x.

The points A (2, 0), B(9, 1), C (11, 6) and D (4, 4) are the vertices of a quadrilateral ABCD. Determine whether ABCD is a rhombus or not.

Find the coordinates of the centre of the circle inscribed in a triangle whose vertices are (-36, 7), (20, 7) and (0, -8).

The base of an equilateral triangle with side 2a lies along the y-axis such that the mid-point of the base is at the origin. Find the vertices of the triangle.

Find the distance between P(x₁, y₁) and Q(x₂, y₂) when (i) PQ is parallel to the y-axis (ii) PQ is parallel to the x-axis.

Find a point on the x-axis, which is equidistant from the point (7, 6) and (3, 4).

Find the locus of a point equidistant from the point (2, 4) and the y-axis.

Find the equation of the locus of a point which moves such that the ratio of its distance from (2, 0) and (1, 3) is 5 : 4.

A point moves as so that the difference of its distances from (ae, 0) and (-ae, 0) is 2a, prove that the equation to its locus is , where b² = a²(e² – 1).

Find the locus of a point such that the sum of its distances from (0, 2) and (0, -2) is 6.

Find the locus of a point which is equidistant from (1, 3) and x-axis.

Find the locus of a point which moves such that its distance from the origin is three times is distance from x-axis.

A(5, 3), B(3, -2) are two fixed points, find the equation to the locus of a point P which moves so that the area of the triangle PAB is 9 units.

Find the locus of a point such that the line segments having end points (2, 0) and (-2, 0) subtend a right angle at that point.

If A (-1, 1) and B (2, 3) are two fixed points, find the locus of a point P so that the area d ΔPAB = 8 sq. units.

A rod of length l slides between the two perpendicular lines. Find the locus of the point on the rod which divides it in the ratio 1 : 2.

Find the locus of the mid-point of the portion of the x cos α + y sin α = p which is intercepted between the axes.

If O is the origin and Q is a variable point on y² = x, Find the locus of the mid-point of OQ.

What does the equation (x – a)² + (y – b)² = r² become when the axes are transferred to parallel axes through the point (a-c, b)?

What does the equation (a – b) (x² + y²) – 2abx = 0 become if the origin is shifted to the point (ab/(a-b), 0) without rotation?

Find what the following equations become when the origin is shifted to the point (1, 1)?

(i) x² + xy – 3x – y + 2 = 0

(ii) x² – y² – 2x + 2y = 0

(iii) xy – x – y + 1 = 0

(iv) xy – y² – x + y = 0

At what point the origin be shifted so that the equation x² + xy – 3x + 2 = 0 does not contain any first-degree term and constant term?

Verify that the area of the triangle with vertices (2, 3), (5, 7) and (-3 -1) remains invariant under the translation of axes when the origin is shifted to the point (-1, 3).

Find, what the following equations become when the origin is shifted to the point (1, 1).

(i) x² + xy – 3y² – y + 2 = 0

(ii) xy – y² – x + y = 0

(iii) xy – x – y + 1 = 0

(iv) x² – y² – 2x + 2y = 0

Find the point to which the origin should be shifted after a translation of axes so that the following equations will have no first degree terms:

(i) y² + x² – 4x – 8y + 3 = 0

(ii) x² + y² – 5x + 2y – 5 = 0

(iii) x² – 12x + 4 = 0

Verify that the area of the triangle with vertices (4, 6), (7, 10) and (1, -2) remains invariant under the translation of axes when the origin is shifted to the point (-2, 1).