Advertisements
Advertisements
प्रश्न
In Q.No. 1, write the distance between the circumcentre and orthocentre of ∆OAB.
Advertisements
उत्तर
The coordinates of circumcentre of a triangle are the point of intersection of perpendicular bisectors of any two sides of the triangle.
Thus, the coordinates of the circumcentre of triangle OAB is \[\left( \frac{a}{2}, \frac{b}{2} \right)\], as shown in the figure.
We know that the orthocentre of a triangle is the intersection of any two altitudes of the triangle.
So, the orthocentre of triangle OAB is the origin O(0, 0).
\[\therefore\] Distance between the circumcentre and orthocentre of ∆OAB = OC
\[\Rightarrow OC = \sqrt{\left( \frac{a}{2} - 0 \right)^2 + \left( \frac{b}{2} - 0 \right)^2} = \frac{\sqrt{a^2 + b^2}}{2}\]
APPEARS IN
संबंधित प्रश्न
The vertices of a triangle ABC are A (0, 0), B (2, −1) and C (9, 2). Find cos B.
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.
Find the coordinates of the centre of the circle inscribed in a triangle whose vertices are (−36, 7), (20, 7) and (0, −8).
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.
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 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.
If O is the origin and Q is a variable point on y2 = x, find the locus of the mid-point of OQ.
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)?
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
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
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.
Write the coordinates of the orthocentre of the triangle formed by points (8, 0), (4, 6) and (0, 0).
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 coordinates of the in-centre of the triangle with vertices at (0, 0), (5, 0) and (0, 12).
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.
