Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Slope of AC = \[\frac{8 - 2}{5 - 1} = \frac{3}{2}\]
The sides AB and AD pass through the point A(1,2) and make an angle of \[{45}^\circ\] with AC whose slope is \[\frac{3}{2}\].
Equations of AB and AD are given by \[y - 2 = \frac{\frac{3}{2} \pm \tan {45}^\circ}{1 \mp \frac{3}{2}\tan {45}^\circ}\left( x - 1 \right)\]
\[\Rightarrow y - 2 = \frac{3 \pm 2}{2 \mp 3}\left( x - 1 \right)\]
\[\Rightarrow y - 2 = - 5\left( x - 1 \right) \text { and } y - 2 = \frac{1}{5}\left( x - 1 \right)\]
\[ \Rightarrow 5x + y - 7 = 0 \text { and } x - 5y + 9 = 0\]
Thus, the equations of AB and AD are \[5x + y - 7 = 0 \text { and } x - 5y + 9 = 0\] respectively.
Since BC is parallel to AD, the equation of BC is \[x - 5y + \lambda = 0\].
This line passes through C (5,8).
\[5 - 40 + \lambda = 0 \Rightarrow \lambda = 35\]
So, the equation of BC is \[x - 5y + 35 = 0\].
Since CD is parallel to AB, the equation of CD is \[5x + y + \lambda = 0\].
This line passes through C (5, 8).
\[25 + 8 + \lambda = 0 \Rightarrow \lambda = - 33\]
So, the equation of CD is \[5x + y - 33 = 0\].
Solving equation of AB and BC, we get B as (0, 7).
Solving equation of AD and CD, we get D as (6, 3).
Hence, the other two vertices are (0, 7) and (6, 3).
APPEARS IN
संबंधित प्रश्न
Without using the Pythagoras theorem, show that the points (4, 4), (3, 5) and (–1, –1) are the vertices of a right angled triangle.
Find the slope of the line, which makes an angle of 30° with the positive direction of y-axis measured anticlockwise.
A line passes through (x1, y1) and (h, k). If slope of the line is m, show that k – y1 = m (h – x1).
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 slope of the lines which make the following angle with the positive direction of x-axis:
\[- \frac{\pi}{4}\]
Find the slope of the lines which make the following angle with the positive direction of x-axis: \[\frac{\pi}{3}\]
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)
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.
If three points A (h, 0), P (a, b) and B (0, k) lie on a line, show that: \[\frac{a}{h} + \frac{b}{k} = 1\].
Find the angle between the X-axis and the line joining the points (3, −1) and (4, −2).
Find the value of x for which the points (x, −1), (2, 1) and (4, 5) are collinear.
Find the equations of the straight lines which cut off an intercept 5 from the y-axis and are equally inclined to the axes.
Show that the perpendicular bisectors of the sides of a triangle are concurrent.
Prove that the straight lines (a + b) x + (a − b ) y = 2ab, (a − b) x + (a + b) y = 2ab and x + y = 0 form an isosceles triangle whose vertical angle is 2 tan−1 \[\left( \frac{a}{b} \right)\].
The angle between the lines 2x − y + 3 = 0 and x + 2y + 3 = 0 is
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
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
The coordinates of the foot of the perpendicular from the point (2, 3) on the line x + y − 11 = 0 are
If m1 and m2 are slopes of lines represented by 6x2 - 5xy + y2 = 0, then (m1)3 + (m2)3 = ?
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 = ______.
Point of the curve y2 = 3(x – 2) at which the normal is parallel to the line 2y + 4x + 5 = 0 is ______.
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 two lines ax + by = c and a′x + b′y = c′ are perpendicular if ______.
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 ______.
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.
Slope of a line which cuts off intercepts of equal lengths on the axes is ______.
The point (4, 1) undergoes the following two successive transformations:
(i) Reflection about the line y = x
(ii) Translation through a distance 2 units along the positive x-axis Then the final coordinates of the point are ______.
The points (3, 4) and (2, – 6) are situated on the ______ of the line 3x – 4y – 8 = 0.
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.
| 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 ______.
