Advertisements
Advertisements
Question
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
Solution
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
RELATED QUESTIONS
The slope of a line is double of the slope of another line. If tangent of the angle between them is `1/3`, find the slopes of the lines.
Find the value of p so that the three lines 3x + y – 2 = 0, px + 2y – 3 = 0 and 2x – y – 3 = 0 may intersect at one point.
Find the slope of the lines which make the following angle with the positive direction of x-axis:
\[\frac{3\pi}{4}\]
Find the slope of the lines which make the following angle with the positive direction of x-axis: \[\frac{\pi}{3}\]
Find the slope of a line passing through the following point:
\[(a t_1^2 , 2 a t_1 ) \text { and } (a t_2^2 , 2 a t_2 )\]
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).
Using the method of slope, show that the following points are collinear A (4, 8), B (5, 12), C (9, 28).
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).
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 angle between the X-axis and the line joining the points (3, −1) and (4, −2).
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.
A quadrilateral has vertices (4, 1), (1, 7), (−6, 0) and (−1, −9). Show that the mid-points of the sides of this quadrilateral form a parallelogram.
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.
Find the coordinates of the orthocentre of the triangle whose vertices are (−1, 3), (2, −1) and (0, 0).
Show that the perpendicular bisectors of the sides of a triangle are concurrent.
Find the equations of the altitudes of a ∆ ABC whose vertices are A (1, 4), B (−3, 2) and C (−5, −3).
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:
3x − y + 5 = 0 and x − 3y + 1 = 0
Find the angles between the following pair of straight lines:
3x + 4y − 7 = 0 and 4x − 3y + 5 = 0
If θ is the angle which the straight line joining the points (x1, y1) and (x2, y2) subtends at the origin, prove that \[\tan \theta = \frac{x_2 y_1 - x_1 y_2}{x_1 x_2 + y_1 y_2}\text { and } \cos \theta = \frac{x_1 x_2 + y_1 y_2}{\sqrt{{x_1}^2 + {y_1}^2}\sqrt{{x_2}^2 + {y_2}^2}}\].
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 angle between the lines 2x − y + 3 = 0 and x + 2y + 3 = 0 is
The coordinates of the foot of the perpendicular from the point (2, 3) on the line x + y − 11 = 0 are
The reflection of the point (4, −13) about the line 5x + y + 6 = 0 is
The line passing through (– 2, 0) and (1, 3) makes an angle of ______ with X-axis.
A ray of light coming from the point (1, 2) is reflected at a point A on the x-axis and then passes through the point (5, 3). Find the coordinates of the point A.
If one diagonal of a square is along the line 8x – 15y = 0 and one of its vertex is at (1, 2), then find the equation of sides of the square passing through this vertex.
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).
The equation of the straight line passing through the point (3, 2) and perpendicular to the line y = x is ______.
The line `x/a + y/b` = 1 moves in such a way that `1/a^2 + 1/b^2 = 1/c^2`, where c is a constant. The locus of the foot of the perpendicular from the origin on the given line is x2 + y2 = c2.
The line which passes through the origin and intersect the two lines `(x - 1)/2 = (y + 3)/4 = (z - 5)/3, (x - 4)/2 = (y + 3)/3 = (z - 14)/4`, is ______.
The three straight lines ax + by = c, bx + cy = a and cx + ay = b are collinear, if ______.
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 ______.
