Advertisements
Advertisements
Question
Find what the following equation become when the origin is shifted to the point (1, 1).
xy − y2 − x + y = 0
Advertisements
Solution
Substituting \[x = X + 1, y = Y + 1\] in the given equation, we get:
\[\left( X + 1 \right)\left( Y + 1 \right) - \left( Y + 1 \right)^2 - \left( X + 1 \right) + \left( Y + 1 \right) = 0\]
\[ \Rightarrow XY + X + Y + 1 - Y^2 - 1 - 2Y - X - 1 + Y + 1 = 0\]
\[ \Rightarrow XY - Y^2 = 0\]
Hence, the transformed equation is \[xy - y^2 = 0\]
APPEARS IN
RELATED QUESTIONS
The vertices of a triangle ABC are A (0, 0), B (2, −1) and C (9, 2). Find cos B.
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).
Find a point on the x-axis, which is equidistant from the points (7, 6) and (3, 4).
Find the locus of a point equidistant from the point (2, 4) and the y-axis.
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 \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\]
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 the x-axis.
Find the locus of a point which moves such that its distance from the origin is three times its distance from the x-axis.
Find the locus of a point such that the line segments with end points (2, 0) and (−2, 0) subtend a right angle at that point.
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.
Find the locus of the mid-point of the portion of the line x cos α + y sin α = p which is intercepted between the axes.
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 \[\left( \frac{ab}{a - b}, 0 \right)\] without rotation?
Find what the following equation become when the origin is shifted to the point (1, 1).
x2 − y2 − 2x +2y = 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 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
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).
In Q.No. 1, write the distance between the circumcentre and orthocentre of ∆OAB.
Write the coordinates of the orthocentre of the triangle formed by points (8, 0), (4, 6) and (0, 0).
Three vertices of a parallelogram, taken in order, are (−1, −6), (2, −5) and (7, 2). Write the coordinates of its fourth vertex.
If the points (a, 0), (at12, 2at1) and (at22, 2at2) are collinear, write the value of t1 t2.
If the coordinates of the sides AB and AC of ∆ABC are (3, 5) and (−3, −3), respectively, then write the length of side BC.
Write the coordinates of the circumcentre of a triangle whose centroid and orthocentre are at (3, 3) and (−3, 5), respectively.
Write the area of the triangle with vertices at (a, b + c), (b, c + a) and (c, a + b).
