Advertisements
Advertisements
प्रश्न
Find the orthocentre of the triangle the equations of whose sides are x + y = 1, 2x + 3y = 6 and 4x − y + 4 = 0.
उत्तर
The given lines are as follows:
x + y = 1 ... (1)
2x + 3y = 6 ... (2)
4x − y + 4 = 0 ... (3)
In triangle ABC, let equations (1), (2) and (3) represent the sides AB, BC and CA, respectively.
Solving (1) and (2):
x = −3, y = 4
Thus, AB and BC intersect at B (−3, 4).
Solving (1) and (3):
x = \[- \frac{3}{5}\] , y = \[\frac{8}{5}\]
Thus, AB and CA intersect at
\[A \left( - \frac{3}{5}, \frac{8}{5} \right)\].
Let AD and BE be the altitudes.
\[AD \perp BC \text { and }BE \perp AC\]
\[\therefore\] Slope of AD \[\times\] Slope of BC = −1
and Slope of BE \[\times\] Slope of AC = −1
Here, slope of BC = slope of the line (2) = \[- \frac{2}{3}\] and slope of AC = slope of the line (3) = 4
\[\therefore \text { Slope of AD } \times \left( - \frac{2}{3} \right) = - 1 \text { and slope of BE } \times 4 = - 1\]
\[ \Rightarrow \text { Slope of AD } = \frac{3}{2} \text { and slope of BE } = - \frac{1}{4}\]
The equation of the altitude AD passing through \[A \left( - \frac{3}{5}, \frac{8}{5} \right)\] and having slope \[\frac{3}{2}\] is \[y - \frac{8}{5} = \frac{3}{2}\left( x + \frac{3}{5} \right)\]
\[\Rightarrow 3x - 2y + 5 = 0\] ... (4)
The equation of the altitude BE passing through B (−3, 4) and having slope \[- \frac{1}{4}\] is \[y - 4 = - \frac{1}{4}\left( x + 3 \right)\]
\[\Rightarrow x + 4y - 13 = 0\] ... (5)
Solving (4) and (5), we get
\[\left( \frac{3}{7}, \frac{22}{7} \right)\] as the orthocentre of the triangle.
APPEARS IN
संबंधित प्रश्न
Reduce the following equation into intercept form and find their intercepts on the axes.
3y + 2 = 0
Find equation of the line parallel to the line 3x – 4y + 2 = 0 and passing through the point (–2, 3).
Prove that the line through the point (x1, y1) and parallel to the line Ax + By + C = 0 is A (x –x1) + B (y – y1) = 0.
Find the coordinates of the foot of perpendicular from the point (–1, 3) to the line 3x – 4y – 16 = 0.
Find equation of the line which is equidistant from parallel lines 9x + 6y – 7 = 0 and 3x + 2y + 6 = 0.
A person standing at the junction (crossing) of two straight paths represented by the equations 2x – 3y+ 4 = 0 and 3x + 4y – 5 = 0 wants to reach the path whose equation is 6x – 7y + 8 = 0 in the least time. Find equation of the path that he should follow.
Find the equation of a line which is equidistant from the lines x = − 2 and x = 6.
Find the equation of a line making an angle of 150° with the x-axis and cutting off an intercept 2 from y-axis.
Find the equation of a line that has y-intercept −4 and is parallel to the line joining (2, −5) and (1, 2).
Find the equation of the right bisector of the line segment joining the points (3, 4) and (−1, 2).
Find the equation of the right bisector of the line segment joining the points A (1, 0) and B (2, 3).
Find the equation of the side BC of the triangle ABC whose vertices are (−1, −2), (0, 1) and (2, 0) respectively. Also, find the equation of the median through (−1, −2).
Find the equations of the diagonals of the square formed by the lines x = 0, y = 0, x = 1 and y =1.
Find the equation of the line on which the length of the perpendicular segment from the origin to the line is 4 and the inclination of the perpendicular segment with the positive direction of x-axis is 30°.
Find the equation of the straight line upon which the length of the perpendicular from the origin is 2 and the slope of this perpendicular is \[\frac{5}{12}\].
If the straight line through the point P (3, 4) makes an angle π/6 with the x-axis and meets the line 12x + 5y + 10 = 0 at Q, find the length PQ.
Reduce the equation \[\sqrt{3}\] x + y + 2 = 0 to slope-intercept form and find slope and y-intercept;
Find the point of intersection of the following pairs of lines:
bx + ay = ab and ax + by = ab.
Find the coordinates of the vertices of a triangle, the equations of whose sides are
y (t1 + t2) = 2x + 2a t1t2, y (t2 + t3) = 2x + 2a t2t3 and, y (t3 + t1) = 2x + 2a t1t3.
For what value of λ are the three lines 2x − 5y + 3 = 0, 5x − 9y + λ = 0 and x − 2y + 1 = 0 concurrent?
If the three lines ax + a2y + 1 = 0, bx + b2y + 1 = 0 and cx + c2y + 1 = 0 are concurrent, show that at least two of three constants a, b, c are equal.
Find the equation of the straight line which has y-intercept equal to \[\frac{4}{3}\] and is perpendicular to 3x − 4y + 11 = 0.
Find the values of the parameter a so that the point (a, 2) is an interior point of the triangle formed by the lines x + y − 4 = 0, 3x − 7y − 8 = 0 and 4x − y − 31 = 0.
Determine whether the point (−3, 2) lies inside or outside the triangle whose sides are given by the equations x + y − 4 = 0, 3x − 7y + 8 = 0, 4x − y − 31 = 0 .
If a ≠ b ≠ c, write the condition for which the equations (b − c) x + (c − a) y + (a − b) = 0 and (b3 − c3) x + (c3 − a3) y + (a3 − b3) = 0 represent the same line.
The point which divides the join of (1, 2) and (3, 4) externally in the ratio 1 : 1
The equations of the sides AB, BC and CA of ∆ ABC are y − x = 2, x + 2y = 1 and 3x + y + 5 = 0 respectively. The equation of the altitude through B is
Two vertices of a triangle are (−2, −1) and (3, 2) and third vertex lies on the line x + y = 5. If the area of the triangle is 4 square units, then the third vertex is
The centroid of a triangle is (2, 7) and two of its vertices are (4, 8) and (−2, 6). The third vertex is
A point equidistant from the line 4x + 3y + 10 = 0, 5x − 12y + 26 = 0 and 7x+ 24y − 50 = 0 is
The inclination of the line x – y + 3 = 0 with the positive direction of x-axis is ______.
If the line `x/"a" + y/"b"` = 1 passes through the points (2, –3) and (4, –5), then (a, b) is ______.
Locus of the mid-points of the portion of the line x sin θ + y cos θ = p intercepted between the axes is ______.
Reduce the following equation into slope-intercept form and find their slopes and the y-intercepts.
6x + 3y – 5 = 0
