Advertisements
Advertisements
Question
Write the coordinates of the orthocentre of the triangle formed by points (8, 0), (4, 6) and (0, 0).
Advertisements
Solution
The intersection point of three altitudes of a triangle is called orthocentre.
In the figure, two altitudes ON and BM of ∆OAB are shown.
Slope of AB = \[\frac{6 - 0}{4 - 8} = - \frac{3}{2}\]
\[\therefore\] Slope of ON \[= \frac{2}{3} \left( \because \text{ Product of slopes }= - 1 \right)\]
Equation of ON:
\[\left( y - 0 \right) = \frac{2}{3}\left( x - 0 \right)\]
\[y = \frac{2}{3}x\] ... (1)
Equation of BM:
x = 4 ... (2)
On solving equations (1) and (2), we get
APPEARS IN
RELATED QUESTIONS
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
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.
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).
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 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.
If A (−1, 1) and B (2, 3) are two fixed points, find the locus of a point P, so that the area of ∆PAB = 8 sq. units.
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 (x − a)2 + (y − b)2 = r2 become when the axes are transferred to parallel axes through the point (a − c, b)?
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 + 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
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).
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: y2 + x2 − 4x − 8y + 3 = 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.
In Q.No. 1, write the distance between the circumcentre and orthocentre of ∆OAB.
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.
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.
Write the area of the triangle with vertices at (a, b + c), (b, c + a) and (c, a + b).
