Advertisements
Advertisements
Question
Write the coordinates of the image of the point (3, 8) in the line x + 3y − 7 = 0.
Advertisements
Solution
Let the given point be A(3,8) and its image in the line x + 3y − 7 = 0 is B(h,k).
The midpoint of AB is \[\left( \frac{3 + h}{2}, \frac{8 + k}{2} \right)\] that lies on the line x + 3y − 7 = 0.
\[\therefore \frac{3 + h}{2} + 3 \times \frac{8 + k}{2} - 7 = 0\]
\[h + 3k + 13 = 0\] ... (1)
AB and the line x + 3y − 7 = 0 are perpendicular.
\[\therefore\text { Slope of AB } \times \text { Slope of the line } = - 1\]
\[ \Rightarrow \frac{k - 8}{h - 3} \times \frac{- 1}{3} = - 1\]
\[\Rightarrow 3h - k - 1 = 0\] ... (2)
Solving (1) and (2), we get:
(h, k) = (−1, −4)
Hence, the image of the point (3,8) in the line x + 3y − 7 = 0 is (−1,−4).
APPEARS IN
RELATED QUESTIONS
The base of an equilateral triangle with side 2a lies along they y-axis such that the mid point of the base is at the origin. Find vertices of the triangle.
Without using distance formula, show that points (–2, –1), (4, 0), (3, 3) and (–3, 2) are vertices of a parallelogram.
A line passes through (x1, y1) and (h, k). If slope of the line is m, show that k – y1 = m (h – x1).
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 slope of the lines which make the following angle with the positive direction of x-axis:
\[\frac{2\pi}{3}\]
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 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.
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.
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 angle between X-axis and the line joining the points (3, −1) and (4, −2).
Find the equation of a straight line with slope 2 and y-intercept 3 .
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 perpendicular to the line segment joining (4, 3) and (−1, 1) if it cuts off an intercept −3 from y-axis.
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 equations of the straight lines which cut off an intercept 5 from the y-axis and are equally inclined to the axes.
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 acute angle between the lines 2x − y + 3 = 0 and x + y + 2 = 0.
Find the angle between the line joining the points (2, 0), (0, 3) and the line x + y = 1.
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 \[\frac{x}{a} + \frac{y}{b} = 1 \text { and } \frac{x}{a} - \frac{y}{b} = 1\text { is } \frac{2ab}{a^2 - b^2}\].
If two opposite vertices of a square are (1, 2) and (5, 8), find the coordinates of its other two vertices and the equations of its sides.
The reflection of the point (4, −13) about the line 5x + y + 6 = 0 is
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 the slopes of the lines given by the equation ax2 + 2hxy + by2 = 0 are in the ratio 5 : 3, then the ratio h2 : ab = ______.
The line passing through (– 2, 0) and (1, 3) makes an angle of ______ with X-axis.
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.
The equation of the line passing through (1, 2) and perpendicular to x + y + 7 = 0 is ______.
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.
The coordinates of the foot of perpendiculars from the point (2, 3) on the line y = 3x + 4 is given by ______.
Equation of the line passing through (1, 2) and parallel to the line y = 3x – 1 is ______.
Equations of the lines through the point (3, 2) and making an angle of 45° with the line x – 2y = 3 are ______.
If the vertices of a triangle have integral coordinates, then the triangle can not be equilateral.
The points A(– 2, 1), B(0, 5), C(– 1, 2) are collinear.
The equation of the line through the intersection of the lines 2x – 3y = 0 and 4x – 5y = 2 and
| Column C1 | Column C2 |
| (a) Through the point (2, 1) is | (i) 2x – y = 4 |
| (b) Perpendicular to the line (ii) x + y – 5 = 0 x + 2y + 1 = 0 is |
(ii) x + y – 5 = 0 |
| (c) Parallel to the line (iii) x – y –1 = 0 3x – 4y + 5 = 0 is |
(iii) x – y –1 = 0 |
| (d) Equally inclined to the axes is | (iv) 3x – 4y – 1 = 0 |
