Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Let (x1, y1) be the coordinates of the given point P and m be the slope of the line.
∴ Equation of the line is y – y1 = m(x – x1) .......(i)
Given points are A(2, 0), B(0, 2) and C(1, 1).
Perpendicular distance from A(2, 0) to the line (i) d1 (say)
d1 = `(0 - y_1 - m(2 - x_1))/sqrt(1 + m^2)`
Perpendicular distance from B(0, 2) d2 (say)
d2 = `(2 - y_1 - m(0 - x_1))/sqrt(1 + m^2)`
Similarly, perpendicular distance from C(1, 1) d3 (say)
d3 = `(1 - y_1 - m(1 - x_1))/sqrt(1 + m^2)`
We have d1 + d2 + d3 = 0
∴ `(0 - y_1 - m(2 - x_1))/sqrt(1 + m^2) + (2 - y_1 - m(0 - x_1))/sqrt(1 + m^2) + (1 - y_1 - m(1 - x_1))/sqrt(1 + m^2)` = 0
⇒ – y1 – 2m + mx1 + 2 – y1 + mx1 + 1 – y1 – m + mx1 = 0
⇒ 3mx1 – 3y1 – 3m + 3 = 0
⇒ mx1 – y1 – m + 1 = 0
Since the point (1, 1) satisfies the above equation.
Hence, the point (1, 1) lies on the line.
APPEARS IN
संबंधित प्रश्न
Find the slope of the line, which makes an angle of 30° with the positive direction of y-axis measured anticlockwise.
Find the angle between the x-axis and the line joining the points (3, –1) and (4, –2).
A line passes through (x1, y1) and (h, k). If slope of the line is m, show that k – y1 = m (h – x1).
If three point (h, 0), (a, b) and (0, k) lie on a line, show that `q/h + b/k = 1`
Consider the given population and year graph. Find the slope of the line AB and using it, find what will be the population in the year 2010?
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 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)
State whether the two lines in each of the following is parallel, perpendicular or neither.
Through (6, 3) and (1, 1); through (−2, 5) and (2, −5)
State whether the two lines in each of the following is parallel, perpendicular or neither.
Through (3, 15) and (16, 6); through (−5, 3) and (8, 2).
Find the slope of a line (i) which bisects the first quadrant angle (ii) which makes an angle of 30° with the positive direction of y-axis measured anticlockwise.
Using the method of slope, show that the following points are collinear A (16, − 18), B (3, −6), C (−10, 6) .
What is the value of y so that the line through (3, y) and (2, 7) is parallel to the line through (−1, 4) and (0, 6)?
Show that the line joining (2, −3) and (−5, 1) is parallel to the line joining (7, −1) and (0, 3).
Without using Pythagoras theorem, show that the points A (0, 4), B (1, 2) and C (3, 3) are the vertices of a right angled triangle.
Find the value of x for which the points (x, −1), (2, 1) and (4, 5) are collinear.
By using the concept of slope, show that the points (−2, −1), (4, 0), (3, 3) and (−3, 2) are the vertices of a parallelogram.
Find the equation of a straight line with slope 2 and y-intercept 3 .
Find the equation of a line which is perpendicular to the line joining (4, 2) and (3, 5) and cuts off an intercept of length 3 on y-axis.
Find the coordinates of the orthocentre of the triangle whose vertices are (−1, 3), (2, −1) and (0, 0).
Find the equation of the right bisector of the line segment joining the points (3, 4) and (−1, 2).
If θ is the angle which the straight line joining the points (x1, y1) and (x2, y2) subtends at the origin, prove that \[\tan \theta = \frac{x_2 y_1 - x_1 y_2}{x_1 x_2 + y_1 y_2}\text { and } \cos \theta = \frac{x_1 x_2 + y_1 y_2}{\sqrt{{x_1}^2 + {y_1}^2}\sqrt{{x_2}^2 + {y_2}^2}}\].
Find the tangent of the angle between the lines which have intercepts 3, 4 and 1, 8 on the axes respectively.
Show that the line a2x + ay + 1 = 0 is perpendicular to the line x − ay = 1 for all non-zero real values of a.
Show that the tangent of an angle between the lines \[\frac{x}{a} + \frac{y}{b} = 1 \text { and } \frac{x}{a} - \frac{y}{b} = 1\text { is } \frac{2ab}{a^2 - b^2}\].
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
If m1 and m2 are slopes of lines represented by 6x2 - 5xy + y2 = 0, then (m1)3 + (m2)3 = ?
If the slope of a line passing through the point A(3, 2) is `3/4`, then find points on the line which are 5 units away from the point A.
The two lines ax + by = c and a′x + b′y = c′ are perpendicular if ______.
Find the equation of the line passing through the point (5, 2) and perpendicular to the line joining the points (2, 3) and (3, – 1).
Show that the tangent of an angle between the lines `x/a + y/b` = 1 and `x/a - y/b` = 1 is `(2ab)/(a^2 - b^2)`
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.
Find the equation of one of the sides of an isosceles right angled triangle whose hypotenuse is given by 3x + 4y = 4 and the opposite vertex of the hypotenuse is (2, 2).
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.
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.
The points A(– 2, 1), B(0, 5), C(– 1, 2) are collinear.
