Advertisements
Advertisements
Question
The distance between the orthocentre and circumcentre of the triangle with vertices (1, 2), (2, 1) and \[\left( \frac{3 + \sqrt{3}}{2}, \frac{3 + \sqrt{3}}{2} \right)\] is
Options
0
\[\sqrt{2}\]
\[3 + \sqrt{3}\]
none of these
Advertisements
Solution
0
Let A(1, 2), B(2, 1) and C \[\left( \frac{3 + \sqrt{3}}{2}, \frac{3 + \sqrt{3}}{2} \right)\] be the given points.
\[\therefore \text { AB } = \sqrt{\left( 2 - 1 \right)^2 + \left( 1 - 2 \right)^2}\]
\[ = \sqrt{2}\]
\[\text { BC } = \sqrt{\left( \frac{3 + \sqrt{3}}{2} - 2 \right)^2 + \left( \frac{3 + \sqrt{3}}{2} - 1 \right)^2}\]
\[ = \sqrt{2}\]
\[\text { AC }= \sqrt{\left( \frac{3 + \sqrt{3}}{2} - 1 \right)^2 + \left( \frac{3 + \sqrt{3}}{2} - 2 \right)^2}\]
\[ = \sqrt{2}\]
Thus, ABC is an equilateral triangle.
We know that the orthocentre and the circumcentre of an equilateral triangle are same.
So, the distance between the the orthocentre and the circumcentre of the triangle
with vertices (1, 2), (2, 1) and \[\left( \frac{3 + \sqrt{3}}{2}, \frac{3 + \sqrt{3}}{2} \right)\] is 0.
APPEARS IN
RELATED QUESTIONS
If the lines `(x-1)/2=(y+1)/3=(z-1)/4 ` and `(x-3)/1=(y-k)/2=z/1` intersect each other then find value of k
Find the points on the x-axis, whose distances from the `x/3 +y/4 = 1` are 4 units.
Find the distance between parallel lines:
15x + 8y – 34 = 0 and 15x + 8y + 31 = 0
What are the points on the y-axis whose distance from the line `x/3 + y/4 = 1` is 4 units.
Find the equation of the line parallel to y-axis and drawn through the point of intersection of the lines x– 7y + 5 = 0 and 3x + y = 0.
Find the direction in which a straight line must be drawn through the point (–1, 2) so that its point of intersection with the line x + y = 4 may be at a distance of 3 units from this point.
A ray of light passing through the point (1, 2) reflects on the x-axis at point A and the reflected ray passes through the point (5, 3). Find the coordinates of A.
Find the co-ordinates of the point, which divides the line segment joining the points A(2, − 6, 8) and B(− 1, 3, − 4) externally in the ratio 1 : 3.
Prove that the line y − x + 2 = 0 divides the join of points (3, −1) and (8, 9) in the ratio 2 : 3.
Find the equation of the line whose perpendicular distance from the origin is 4 units and the angle which the normal makes with the positive direction of x-axis is 15°.
Find the distance of the point (2, 3) from the line 2x − 3y + 9 = 0 measured along a line making an angle of 45° with the x-axis.
Find the distance of the point (3, 5) from the line 2x + 3y = 14 measured parallel to the line x − 2y = 1.
Find the distance of the line 2x + y = 3 from the point (−1, −3) in the direction of the line whose slope is 1.
Show that the perpendiculars let fall from any point on the straight line 2x + 11y − 5 = 0 upon the two straight lines 24x + 7y = 20 and 4x − 3y − 2 = 0 are equal to each other.
Show that the product of perpendiculars on the line \[\frac{x}{a} \cos \theta + \frac{y}{b} \sin \theta = 1\] from the points \[( \pm \sqrt{a^2 - b^2}, 0) \text { is }b^2 .\]
Show that the path of a moving point such that its distances from two lines 3x − 2y = 5 and 3x + 2y = 5 are equal is a straight line.
If the length of the perpendicular from the point (1, 1) to the line ax − by + c = 0 be unity, show that \[\frac{1}{c} + \frac{1}{a} - \frac{1}{b} = \frac{c}{2ab}\] .
Determine the distance between the pair of parallel lines:
4x − 3y − 9 = 0 and 4x − 3y − 24 = 0
The equations of two sides of a square are 5x − 12y − 65 = 0 and 5x − 12y + 26 = 0. Find the area of the square.
Prove that the lines 2x + 3y = 19 and 2x + 3y + 7 = 0 are equidistant from the line 2x + 3y= 6.
Write the distance between the lines 4x + 3y − 11 = 0 and 8x + 6y − 15 = 0.
The line segment joining the points (1, 2) and (−2, 1) is divided by the line 3x + 4y = 7 in the ratio ______.
A plane passes through (1, - 2, 1) and is perpendicular to two planes 2x - 2y + z = 0 and x - y + 2z = 4. The distance of the plane from the point (1, 2, 2) is ______.
The shortest distance between the lines
`bar"r" = (hat"i" + 2hat"j" + hat"k") + lambda (hat"i" - hat"j" + hat"k")` and
`bar"r" = (2hat"i" - hat"j" - hat"k") + mu(2hat"i" + hat"j" + 2hat"k")` is
If the tangent to the curve y = 3x2 - 2x + 1 at a point Pis parallel toy = 4x + 3, the co-ordinates of P are
If P(α, β) be a point on the line 3x + y = 0 such that the point P and the point Q(1, 1) lie on either side of the line 3x = 4y + 8, then _______.
Find the distance between the lines 3x + 4y = 9 and 6x + 8y = 15.
Show that the locus of the mid-point of the distance between the axes of the variable line x cosα + y sinα = p is `1/x^2 + 1/y^2 = 4/p^2` where p is a constant.
A point moves such that its distance from the point (4, 0) is half that of its distance from the line x = 16. The locus of the point is ______.
A point equidistant from the lines 4x + 3y + 10 = 0, 5x – 12y + 26 = 0 and 7x + 24y – 50 = 0 is ______.
The ratio in which the line 3x + 4y + 2 = 0 divides the distance between the lines 3x + 4y + 5 = 0 and 3x + 4y – 5 = 0 is ______.
The distance of the point (2, – 3, 1) from the line `(x + 1)/2 = (y - 3)/3 = (z + 1)/-1` is ______.
The distance of the point (-3, 2, 3) from the line passing through (4, 6, -2) and having direction ratios -1, 2, 3 is ______units.
The distance between the parallel lines 3x − 4y + 7 = 0 and 3x − 4y + 5 = 0 is `a/b`. Value of a + b is ______.
