R.D. Sharma solutions for माठेमटिक्स १ [इंग्रजी] इयत्ता ११ chapter 20 - Brief review of cartesian system of rectangular co-ordinates [Latest edition]

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

Four points A (6, 3), B (−3, 5), C (4, −2) and D (x, 3x) are given in such a way that \[\frac{\Delta DBC}{\Delta ABC} = \frac{1}{2}\]. 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.

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.

A point moves so that the difference of its distances from (ae, 0) and (−ae, 0) is 2a. Prove that the equation to its locus is \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\]

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

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

What does the equation (a − b) (x² + y²) −2abx = 0 become if the origin is shifted to the point \[\left( \frac{ab}{a - b}, 0 \right)\] without rotation?

Find what the following equation become when the origin is shifted to the point (1, 1).
x² + xy − 3x − y + 2 = 0

Find what the following equation become when the origin is shifted to the point (1, 1).
 x² − y² − 2x +2y = 0

Find what the following equation become when the origin is shifted to the point (1, 1).
xy − x − y + 1 = 0

Find what the following equation become when the origin is shifted to the point (1, 1).
xy − y² − x + y = 0

To what point should the origin be shifted so that the equation x² + xy − 3x − y + 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 equation become when the origin is shifted to the point (1, 1).
x² + xy − 3y² − y + 2 = 0

Find what the following equation become when the origin is shifted to the point (1, 1).
xy − y² − x + y = 0

Find what the following equation become when the origin is shifted to the point (1, 1).
 xy − x − y + 1 = 0

Find what the following equation become when the origin is shifted to the point (1, 1).
x² − y² − 2x + 2y = 0

Find the point to which the origin should be shifted after a translation of axes so that the following equation will have no first degree terms:  y² + x² − 4x − 8y + 3 = 0

Find the point to which the origin should be shifted after a translation of axes so that the following equation will have no first degree terms: x² + y² − 5x + 2y − 5 = 0

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

The vertices of a triangle are O (0, 0), A (a, 0) and B (0, b). Write the coordinates of its circumcentre.

If the points (1, −1), (2, −1) and (4, −3) are the mid-points of the sides of a triangle, then write the coordinates of its centroid.