Advertisements
Advertisements
Question
Write the coordinates of the orthocentre of the triangle formed by the lines x2 − y2 = 0 and x + 6y = 18.
Solution
The equation x2 − y2 = 0 represents a pair of straight line, which can be written in the following way:
(x + y)(x − y) = 0
So, the lines can be written separately in the following manner:
x + y = 0 ... (1)
x − y = 0 ... (2)
The third line is
x + 6y = 18 ... (3)
Lines (1) and (2) are perpendicular to each other as their slopes are −1 and 1, respectively
⇒ −1 \[\times\] 1 = −1
Therefore, the triangle formed by the lines (1), (2) and (3) is a right-angled triangle.
Thus, the orthocentre of the triangle formed by the given lines is the intersection of x + y = 0 and x − y = 0, which is (0, 0).
APPEARS IN
RELATED QUESTIONS
The perpendicular from the origin to the line y = mx + c meets it at the point (–1, 2). Find the values of m and c.
If p and q are the lengths of perpendiculars from the origin to the lines x cos θ – y sin θ = k cos 2θ and xsec θ+ y cosec θ = k, respectively, prove that p2 + 4q2 = k2.
Find the equations of the lines, which cut-off intercepts on the axes whose sum and product are 1 and –6, respectively.
Find the equation of the line passing through the point of intersection of the lines 4x + 7y – 3 = 0 and 2x – 3y + 1 = 0 that has equal intercepts on the axes.
Find the lines through the point (0, 2) making angles \[\frac{\pi}{3} \text { and } \frac{2\pi}{3}\] with the x-axis. Also, find the lines parallel to them cutting the y-axis at a distance of 2 units below the origin.
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 \[\frac{5}{12}\].
Find the equation of the straight line which makes a triangle of area \[96\sqrt{3}\] with the axes and perpendicular from the origin to it makes an angle of 30° with Y-axis.
Reduce the equation\[\sqrt{3}\] x + y + 2 = 0 to intercept form and find intercept on the axes.
Put the equation \[\frac{x}{a} + \frac{y}{b} = 1\] to the slope intercept form and find its slope and y-intercept.
Find the point of intersection of the following pairs of lines:
2x − y + 3 = 0 and x + y − 5 = 0
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.
Find the equations of the medians of a triangle, the equations of whose sides are:
3x + 2y + 6 = 0, 2x − 5y + 4 = 0 and x − 3y − 6 = 0
Prove that the lines \[y = \sqrt{3}x + 1, y = 4 \text { and } y = - \sqrt{3}x + 2\] form an equilateral triangle.
Prove that the following sets of three lines are concurrent:
3x − 5y − 11 = 0, 5x + 3y − 7 = 0 and x + 2y = 0
Prove that the following sets of three lines are concurrent:
\[\frac{x}{a} + \frac{y}{b} = 1, \frac{x}{b} + \frac{y}{a} = 1\text { and } y = x .\]
Find the conditions that the straight lines y = m1 x + c1, y = m2 x + c2 and y = m3 x + c3 may meet in a point.
If the lines p1 x + q1 y = 1, p2 x + q2 y = 1 and p3 x + q3 y = 1 be concurrent, show that the points (p1, q1), (p2, q2) and (p3, q3) are collinear.
Find the equation of a line which is perpendicular to the line \[\sqrt{3}x - y + 5 = 0\] and which cuts off an intercept of 4 units with the negative direction of y-axis.
Find the equation of the straight line perpendicular to 2x − 3y = 5 and cutting off an intercept 1 on the positive direction of the x-axis.
Find the projection of the point (1, 0) on the line joining the points (−1, 2) and (5, 4).
The equations of perpendicular bisectors of the sides AB and AC of a triangle ABC are x − y + 5 = 0 and x + 2y = 0 respectively. If the point A is (1, −2), find the equation of the line BC.
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.
Write the coordinates of the orthocentre of the triangle formed by the lines xy = 0 and x + y = 1.
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.
If the lines ax + 12y + 1 = 0, bx + 13y + 1 = 0 and cx + 14y + 1 = 0 are concurrent, then a, b, c are in
The figure formed by the lines ax ± by ± c = 0 is
Find the equation of the straight line which passes through the point (1, – 2) and cuts off equal intercepts from axes.
Find the equation of the line which passes through the point (– 4, 3) and the portion of the line intercepted between the axes is divided internally in the ratio 5 : 3 by this point.
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 intercept form and find their intercepts on the axes.
3x + 2y – 12 = 0
Reduce the following equation into intercept form and find their intercepts on the axes.
4x – 3y = 6
