Advertisements
Advertisements
प्रश्न
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?
उत्तर
Let the origin be shifted to (h, k). Then, x = X + h and y = Y + k.
Substituting x = X + h and y = Y + k in the equation x2 + xy − 3x − y + 2 = 0, we get:
\[\left( X + h \right)^2 + \left( X + h \right)\left( Y + k \right) - 3\left( X + h \right) - \left( Y + k \right) + 2 = 0\]
\[ \Rightarrow X^2 + 2hX + h^2 + XY + kX + hY + hk - 3X - 3h - Y - k + 2 = 0\]
\[ \Rightarrow X^2 + XY + X\left( 2h + k - 3 \right) + Y\left( h - 1 \right) + h^2 + hk - 3h - k + 2 = 0\]
For this equation to be free from the first-degree terms and constant term, we must have
\[2h + k - 3 = 0, h - 1 = 0, h^2 + hk - 3h - k + 2 = 0\]
\[ \Rightarrow h = 1, k = 1, h^2 + hk - 3k - h + 2 = 0\]
Also, h =1 and k = 1 satisfy the equation \[h^2 + hk - 3k - h + 2 = 0\]
Hence, the origin should be shifted to the point (1, 1).
APPEARS IN
संबंधित प्रश्न
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, 2). Find cos B.
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 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.
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 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 (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
Find what the following equation become when the origin is shifted to the point (1, 1).
xy − y2 − x + y = 0
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: 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
The vertices of a triangle are O (0, 0), A (a, 0) and B (0, b). Write the coordinates of its circumcentre.
In Q.No. 1, write the distance between the circumcentre and orthocentre of ∆OAB.
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.
