Advertisements
Advertisements
प्रश्न
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 ______.
विकल्प
(4, 3)
(3, 4)
(1, 4)
`7/2, 7/2`
उत्तर
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 (3, 4).
Explanation:
Let the reflection of A(4, 1) in y = x be B(a, b) mid-point of AB
= `((4 + a)/2, (1 + b)/2)` which lies on y = x
⇒ `(4 + "a")/2 = (1 + b)/2`
⇒ 4 + a = 1 + b
⇒ a – b = – 3 .......(i)
The slope of the line y = x is 1 and slope of AB = `(b - 1)/(a - 4)`
∴ `1((b - 1)/(a - 4)) = - 1`
⇒ b – 1 = – a + 4
⇒ a + b = 5 ....(ii)
Solving equation (i) and equation (ii) we get
a = 1 and b = 4
∴ The point after translation is (1 + 2, 4) or (3, 4).
APPEARS IN
संबंधित प्रश्न
Draw a quadrilateral in the Cartesian plane, whose vertices are (–4, 5), (0, 7), (5, –5) and (–4, –2). Also, find its area.
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.
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 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 the lines which make the following angle with the positive direction of x-axis:
\[\frac{2\pi}{3}\]
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 are parallel, perpendicular or neither.
Through (5, 6) and (2, 3); through (9, −2) and (6, −5)
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).
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.
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).
Find the equation of the right bisector of the line segment joining the points (3, 4) and (−1, 2).
Find the image of the point (3, 8) with respect to the line x + 3y = 7 assuming the line to be a plane mirror.
Prove that the points (2, −1), (0, 2), (2, 3) and (4, 0) are the coordinates of the vertices of a parallelogram and find the angle between its diagonals.
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 \[\left( \frac{a}{b} \right)\].
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 line passing through (– 2, 0) and (1, 3) makes an angle of ______ with X-axis.
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 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.
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.
P1, P2 are points on either of the two lines `- sqrt(3) |x|` = 2 at a distance of 5 units from their point of intersection. Find the coordinates of the foot of perpendiculars drawn from P1, P2 on the bisector of the angle between the given lines.
Slope of a line which cuts off intercepts of equal lengths on the axes is ______.
One vertex of the equilateral triangle with centroid at the origin and one side as x + y – 2 = 0 is ______.
If the vertices of a triangle have integral coordinates, then the triangle can not be equilateral.
The three straight lines ax + by = c, bx + cy = a and cx + ay = b are collinear, if ______.
