Advertisements
Advertisements
प्रश्न
One vertex of the equilateral triangle with centroid at the origin and one side as x + y – 2 = 0 is ______.
विकल्प
(–1, –1)
(2, 2)
(–2, –2)
(2, –2)
Advertisements
उत्तर
One vertex of the equilateral triangle with centroid at the origin and one side as x + y – 2 = 0 is (–2, –2).
Explanation:
Let ABC be an equilateral triangle with vertex (x1, y1).
AD ⊥ BC and let (a, b) be the coordinates of D.
Given that the centroid G lies at the origin i.e., (0, 0)
Since, the centroid of a triangle,divides the median in the ratio 1 : 2
So, 0 = `(1 xx x_1 + 2 xx a)/(1 + 2)`
⇒ x1 + 2a = 0 ......(i)
And 0 = `(1 xx y_1 + 2 xx b)/(1 + 2)`
⇒ y1 + 2b = 0 ......(ii)
Equations of BC is given by x + y – 2 = 0 .....(iii)
Point D(a, b) lies on the line x + y – 2 = 0
So a + b – 2 = 0
Slope of equation (iii) is = – 1
And the slope of AG = `(y_1 - 0)/(x_1 - 0) = y_1/x_1`
Since, they are perpendicular to each other
∴ `- 1 xx y_1/x_1` = – 1
⇒ y1 = x1
From eq. (i) and (ii) we get
x1 + 2a = 0
⇒ 2a = – x1
y1 + 2b = 0
⇒ 2b = – y1
∴ a = b
From equation (iv) we get
a + b – 2 = 0
⇒ a + a – 2 = 0
⇒ 2a – 2 = 0
⇒ a = 1 and b = 1 ....[∵ a = b]
∴ x1 = – 2 × 1 = – 2
And y1 = – 2 × 1 = – 2
APPEARS IN
संबंधित प्रश्न
Find the distance between P (x1, y1) and Q (x2, y2) when :
- PQ is parallel to the y-axis,
- PQ is parallel to the x-axis
Find the value of x for which the points (x, –1), (2, 1) and (4, 5) are collinear.
Find the slope of a line passing through the following point:
\[(a t_1^2 , 2 a t_1 ) \text { and } (a t_2^2 , 2 a t_2 )\]
State whether the two lines in each of the following is parallel, perpendicular or neither.
Through (6, 3) and (1, 1); through (−2, 5) and (2, −5)
State whether the two lines in each of the following is parallel, perpendicular or neither.
Through (3, 15) and (16, 6); through (−5, 3) and (8, 2).
What can be said regarding a line if its slope is negative?
Show that the line joining (2, −3) and (−5, 1) is parallel to the line joining (7, −1) and (0, 3).
Show that the line joining (2, −5) and (−2, 5) is perpendicular to the line joining (6, 3) and (1, 1).
Without using Pythagoras theorem, show that the points A (0, 4), B (1, 2) and C (3, 3) are the vertices of a right angled triangle.
If three points A (h, 0), P (a, b) and B (0, k) lie on a line, show that: \[\frac{a}{h} + \frac{b}{k} = 1\].
Consider the following population and year graph:
Find the slope of the line AB and using it, find what will be the population in the year 2010.
Line through the points (−2, 6) and (4, 8) is perpendicular to the line through the points (8, 12) and (x, 24). Find the value of x.
A quadrilateral has vertices (4, 1), (1, 7), (−6, 0) and (−1, −9). Show that the mid-points of the sides of this quadrilateral form a parallelogram.
Find the equations of the bisectors of the angles between the coordinate axes.
Find the equations of the straight lines which cut off an intercept 5 from the y-axis and are equally inclined to the axes.
Find the coordinates of the orthocentre of the triangle whose vertices are (−1, 3), (2, −1) and (0, 0).
Show that the perpendicular bisectors of the sides of a triangle are concurrent.
Find the equation of the right bisector of the line segment joining the points (3, 4) and (−1, 2).
If two opposite vertices of a square are (1, 2) and (5, 8), find the coordinates of its other two vertices and the equations of its sides.
The equation of the line with slope −3/2 and which is concurrent with the lines 4x + 3y − 7 = 0 and 8x + 5y − 1 = 0 is
The reflection of the point (4, −13) about the line 5x + y + 6 = 0 is
The equation of a line passing through the point (7, - 4) and perpendicular to the line passing through the points (2, 3) and (1 , - 2 ) is ______.
A ray of light coming from the point (1, 2) is reflected at a point A on the x-axis and then passes through the point (5, 3). Find the coordinates of the point A.
The reflection of the point (4, – 13) about the line 5x + y + 6 = 0 is ______.
Find the angle between the lines y = `(2 - sqrt(3)) (x + 5)` and y = `(2 + sqrt(3))(x - 7)`
A variable line passes through a fixed point P. The algebraic sum of the perpendiculars drawn from the points (2, 0), (0, 2) and (1, 1) on the line is zero. Find the coordinates of the point P.
P1, P2 are points on either of the two lines `- sqrt(3) |x|` = 2 at a distance of 5 units from their point of intersection. Find the coordinates of the foot of perpendiculars drawn from P1, P2 on the bisector of the angle between the given lines.
Equation of the line passing through (1, 2) and parallel to the line y = 3x – 1 is ______.
The point (4, 1) undergoes the following two successive transformations:
(i) Reflection about the line y = x
(ii) Translation through a distance 2 units along the positive x-axis Then the final coordinates of the point are ______.
Equations of the lines through the point (3, 2) and making an angle of 45° with the line x – 2y = 3 are ______.
The points (3, 4) and (2, – 6) are situated on the ______ of the line 3x – 4y – 8 = 0.
The line `x/a + y/b` = 1 moves in such a way that `1/a^2 + 1/b^2 = 1/c^2`, where c is a constant. The locus of the foot of the perpendicular from the origin on the given line is x2 + y2 = c2.
| Column C1 | Column C2 |
| (a) The coordinates of the points P and Q on the line x + 5y = 13 which are at a distance of 2 units from the line 12x – 5y + 26 = 0 are |
(i) (3, 1), (–7, 11) |
| (b) The coordinates of the point on the line x + y = 4, which are at a unit distance from the line 4x + 3y – 10 = 0 are |
(ii) `(- 1/3, 11/3), (4/3, 7/3)` |
| (c) The coordinates of the point on the line joining A (–2, 5) and B (3, 1) such that AP = PQ = QB are |
(iii) `(1, 12/5), (-3, 16/5)` |
The equation of the line through the intersection of the lines 2x – 3y = 0 and 4x – 5y = 2 and
| Column C1 | Column C2 |
| (a) Through the point (2, 1) is | (i) 2x – y = 4 |
| (b) Perpendicular to the line (ii) x + y – 5 = 0 x + 2y + 1 = 0 is |
(ii) x + y – 5 = 0 |
| (c) Parallel to the line (iii) x – y –1 = 0 3x – 4y + 5 = 0 is |
(iii) x – y –1 = 0 |
| (d) Equally inclined to the axes is | (iv) 3x – 4y – 1 = 0 |
