Advertisements
Advertisements
Question
Find the coordinates of the orthocentre of the triangle whose vertices are (−1, 3), (2, −1) and (0, 0).
Advertisements
Solution
Let A (0, 0), B (−1, 3) and C (2, −1) be the vertices of the triangle ABC.
Let AD and BE be the altitudes.
\[AD \perp BC\] and \[BE \perp AC\]
\[\therefore\] Slope of AD \[\times\] Slope of BC = −1
Slope of BE \[\times\] Slope of AC = −1
Here, slope of BC = \[\frac{- 1 - 3}{2 + 1} = - \frac{4}{3}\] and slope of AC = \[\frac{- 1 - 0}{2 - 0} = - \frac{1}{2}\]
\[\therefore \text { Slope of AD } \times \left( - \frac{4}{3} \right) = - \text { 1 and slope of BE } \times \left( - \frac{1}{2} \right) = - 1 \]
\[ \Rightarrow \text { Slope of AD } = \frac{3}{4}\text { and slope of BE } = 2\]
The equation of the altitude AD passing through A (0, 0) and having slope \[\frac{3}{4}\] is
\[y - 0 = \frac{3}{4}\left( x - 0 \right)\]
\[ \Rightarrow y = \frac{3}{4}x . . . . (1)\]
The equation of the altitude BE passing through B (−1, 3) and having slope 2 is
\[y - 3 = 2\left( x + 1 \right)\]
\[ \Rightarrow 2x - y + 5 = 0 . . . . (2)\]
Solving (1) and (2):
x = − 4, y = − 3
Hence, the coordinates of the orthocentre is (−4, −3).
APPEARS IN
RELATED QUESTIONS
Find the slope of a line, which passes through the origin, and the mid-point of the line segment joining the points P (0, –4) and B (8, 0).
Without using distance formula, show that points (–2, –1), (4, 0), (3, 3) and (–3, 2) are vertices of a parallelogram.
Find the angle between the x-axis and the line joining the points (3, –1) and (4, –2).
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 equation of the lines through the point (3, 2) which make an angle of 45° with the line x –2y = 3.
Find the slope of the lines which make the following angle with the positive direction of x-axis:
\[- \frac{\pi}{4}\]
Find the slope of the lines which make the following angle with the positive direction of x-axis:
\[\frac{2\pi}{3}\]
Find the slope of a line passing through the following point:
\[(a t_1^2 , 2 a t_1 ) \text { and } (a t_2^2 , 2 a t_2 )\]
Find the slope of a line passing through the following point:
(3, −5), and (1, 2)
Using the method of slope, show that the following points are collinear A (16, − 18), B (3, −6), C (−10, 6) .
Show that the line joining (2, −3) and (−5, 1) is parallel to the line joining (7, −1) and (0, 3).
The slope of a line is double of the slope of another line. If tangents of the angle between them is \[\frac{1}{3}\],find the slopes of the other line.
Find the equation of a straight line with slope 2 and y-intercept 3 .
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.
Show that the perpendicular bisectors of the sides of a triangle are concurrent.
If the image of the point (2, 1) with respect to a line mirror is (5, 2), find the equation of the mirror.
Find the angles between the following pair of straight lines:
x − 4y = 3 and 6x − y = 11
Prove that the straight lines (a + b) x + (a − b ) y = 2ab, (a − b) x + (a + b) y = 2ab and x + y = 0 form an isosceles triangle whose vertical angle is 2 tan−1 \[\left( \frac{a}{b} \right)\].
Find the tangent of the angle between the lines which have intercepts 3, 4 and 1, 8 on the axes respectively.
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 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
Find the equation of a straight line on which length of perpendicular from the origin is four units and the line makes an angle of 120° with the positive direction of x-axis.
If the equation of the base of an equilateral triangle is x + y = 2 and the vertex is (2, – 1), then find the length of the side of the triangle.
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.
Slope of a line which cuts off intercepts of equal lengths on the axes is ______.
The coordinates of the foot of perpendiculars from the point (2, 3) on the line y = 3x + 4 is given by ______.
One vertex of the equilateral triangle with centroid at the origin and one side as x + y – 2 = 0 is ______.
The points (3, 4) and (2, – 6) are situated on the ______ of the line 3x – 4y – 8 = 0.
| 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 |
If the line joining two points A (2, 0) and B (3, 1) is rotated about A in anticlockwise direction through an angle of 15°, then the equation of the line in new position is ______.
