Advertisements
Advertisements
Question
If the image of the point (2, 1) with respect to a line mirror is (5, 2), find the equation of the mirror.
Advertisements
Solution
Let the image of A (2, 1) be B (5, 2). Also, let M be the midpoint of AB.
\[\therefore \text { Coordinates of } M = \left( \frac{2 + 5}{2}, \frac{1 + 2}{2} \right)\]
\[ = \left( \frac{7}{2}, \frac{3}{2} \right)\]
Let CD be the mirror.
Line AB is perpendicular to the mirror CD.
\[\therefore\] Slope of AB \[\times\] Slope of CD = −1
\[\Rightarrow \frac{2 - 1}{5 - 2} \times \text { Slope of CD } = - 1\]
\[ \Rightarrow \text { Slope of CD }= - 3\]
Equation of the mirror CD:
\[y - \frac{3}{2} = - 3\left( x - \frac{7}{2} \right)\]
\[ \Rightarrow 2y - 3 = - 6x + 21\]
\[ \Rightarrow 6x + 2y - 24 = 0\]
\[ \Rightarrow 3x + y - 12 = 0\]
APPEARS IN
RELATED QUESTIONS
Draw a quadrilateral in the Cartesian plane, whose vertices are (–4, 5), (0, 7), (5, –5) and (–4, –2). Also, find its area.
Find the slope of the line, which makes an angle of 30° with the positive direction of y-axis measured anticlockwise.
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.
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 a line drawn perpendicular to the line `x/4 + y/6 = 1`through the point, where it meets the y-axis.
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 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 are parallel, perpendicular or neither.
Through (5, 6) and (2, 3); through (9, −2) and (6, −5)
State whether the two lines in each of the following are parallel, perpendicular or neither.
Through (9, 5) and (−1, 1); through (3, −5) and (8, −3)
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)
What can be said regarding a line if its slope is zero ?
Show that the line joining (2, −5) and (−2, 5) is perpendicular to the line joining (6, 3) and (1, 1).
If three points A (h, 0), P (a, b) and B (0, k) lie on a line, show that: \[\frac{a}{h} + \frac{b}{k} = 1\].
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.
Without using the distance formula, show that points (−2, −1), (4, 0), (3, 3) and (−3, 2) are the vertices of a parallelogram.
Line through the points (−2, 6) and (4, 8) is perpendicular to the line through the points (8, 12) and (x, 24). Find the value of x.
Find the value of x for which the points (x, −1), (2, 1) and (4, 5) are collinear.
Find the equation of a straight line with slope 2 and y-intercept 3 .
Find the equation of a straight line with slope −2 and intersecting the x-axis at a distance of 3 units to the left of origin.
Find the equations of the bisectors of the angles between the coordinate axes.
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 equation of the right bisector of the line segment joining the points (3, 4) and (−1, 2).
The line through (h, 3) and (4, 1) intersects the line 7x − 9y − 19 = 0 at right angle. Find the value of h.
Find the image of the point (3, 8) with respect to the line x + 3y = 7 assuming the line to be a plane mirror.
Find the angles between the following pair of straight lines:
3x + 4y − 7 = 0 and 4x − 3y + 5 = 0
The medians AD and BE of a triangle with vertices A (0, b), B (0, 0) and C (a, 0) are perpendicular to each other, if
Find k, if the slope of one of the lines given by kx2 + 8xy + y2 = 0 exceeds the slope of the other by 6.
If x + y = k is normal to y2 = 12x, then k is ______.
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.
Find the equation to the straight line passing through the point of intersection of the lines 5x – 6y – 1 = 0 and 3x + 2y + 5 = 0 and perpendicular to the line 3x – 5y + 11 = 0.
The coordinates of the foot of the perpendicular from the point (2, 3) on the line x + y – 11 = 0 are ______.
Find the angle between the lines y = `(2 - sqrt(3)) (x + 5)` and y = `(2 + sqrt(3))(x - 7)`
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.
Slope of a line which cuts off intercepts of equal lengths on the axes is ______.
The tangent of angle between the lines whose intercepts on the axes are a, – b and b, – a, respectively, is ______.
| 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)` |
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). The co-ordinates of the point A is ______.
The lines whose vector equations are `r = 2hati - 3hatj + 7hatk + lambda (2hati + phatj + 5hatk) and r = hati - 2hatj + 3hatk + µ(3hati + phatj + phatk)` are perpendicular for all values of λ and µ if p =
