Advertisements
Advertisements
Question
One vertex of the equilateral triangle with centroid at the origin and one side as x + y – 2 = 0 is ______.
Options
(–1, –1)
(2, 2)
(–2, –2)
(2, –2)
Advertisements
Solution
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
RELATED QUESTIONS
The base of an equilateral triangle with side 2a lies along they y-axis such that the mid point of the base is at the origin. Find vertices of the triangle.
Find a point on the x-axis, which is equidistant from the points (7, 6) and (3, 4).
Find the value of x for which the points (x, –1), (2, 1) and (4, 5) are collinear.
If three point (h, 0), (a, b) and (0, k) lie on a line, show that `q/h + b/k = 1`
Find the equation of a line drawn perpendicular to the line `x/4 + y/6 = 1`through the point, where it meets the y-axis.
Find the slope of the lines which make the following angle with the positive direction of x-axis:
\[\frac{2\pi}{3}\]
State whether the two lines in each of the following are parallel, perpendicular or neither.
Through (5, 6) and (2, 3); through (9, −2) and (6, −5)
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).
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.
Find the angle between X-axis and the line joining the points (3, −1) and (4, −2).
By using the concept of slope, show that the points (−2, −1), (4, 0), (3, 3) and (−3, 2) are the vertices of a parallelogram.
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 equation of the perpendicular to the line segment joining (4, 3) and (−1, 1) if it cuts off an intercept −3 from y-axis.
Find the equation of the strainght line intersecting y-axis at a distance of 2 units above the origin and making an angle of 30° with the positive direction of the x-axis.
The line through (h, 3) and (4, 1) intersects the line 7x − 9y − 19 = 0 at right angle. Find the value of h.
Find the angles between the following pair of straight lines:
3x + y + 12 = 0 and x + 2y − 1 = 0
Show that the tangent of an angle between the lines \[\frac{x}{a} + \frac{y}{b} = 1 \text { and } \frac{x}{a} - \frac{y}{b} = 1\text { is } \frac{2ab}{a^2 - b^2}\].
Write the coordinates of the image of the point (3, 8) in the line x + 3y − 7 = 0.
The acute angle between the medians drawn from the acute angles of a right angled isosceles triangle is
The angle between the lines 2x − y + 3 = 0 and x + 2y + 3 = 0 is
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 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 ______.
If the line joining two points A(2, 0) and B(3, 1) is rotated about A in anticlock wise direction through an angle of 15°. Find the equation of the line in new position.
If the slope of a line passing through the point A(3, 2) is `3/4`, then find points on the line which are 5 units away from the point A.
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.
If p is the length of perpendicular from the origin on the line `x/a + y/b` = 1 and a2, p2, b2 are in A.P, then show that a4 + b4 = 0.
The equation of the straight line passing through the point (3, 2) and perpendicular to the line y = x is ______.
The coordinates of the foot of perpendiculars from the point (2, 3) on the line y = 3x + 4 is given by ______.
The points A(– 2, 1), B(0, 5), C(– 1, 2) are collinear.
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.
