Advertisements
Advertisements
Question
Find the image of the point (2, 1) with respect to the line mirror x + y − 5 = 0.
Solution
Let the image of A (2, 1) be B (a, b). Let M be the midpoint of AB.
\[\therefore \text { Coordinates of M are } \equiv \left( \frac{2 + a}{2}, \frac{1 + b}{2} \right)\]
The point M lies on the line x + y − 5 = 0
\[\therefore \frac{2 + a}{2} + \frac{1 + b}{2} - 5 = 0\]
\[\Rightarrow a + b = 7\] ... (1)
Now, the lines x + y − 5 = 0 and AB are perpendicular.
∴ Slope of AB \[\times\] Slope of CD = −1
\[\Rightarrow \frac{b - 1}{a - 2} \times \left( - 1 \right) = - 1\]
\[ \Rightarrow a - 2 = b - 1\]
⇒ \[a - b = 1\] ... (2)
Adding eq (1) and eq (2):
\[2a = 8\]
\[ \Rightarrow a = 4\]
Now, from equation (1):
\[4 + b = 7\]
\[ \Rightarrow b = 3\]
Hence, the image of the point (2, 1) with respect to the line mirror x + y − 5 = 0 is (4, 3).
APPEARS IN
RELATED QUESTIONS
Find angles between the lines `sqrt3x + y = 1 and x + sqrt3y = 1`.
If p is the length of perpendicular from the origin to the line whose intercepts on the axes are a and b, then show that `1/p^2 = 1/a^2 + 1/b^2`.
If three lines whose equations are y = m1x + c1, y = m2x + c2 and y = m3x + c3 are concurrent, then show that m1(c2 – c3) + m2 (c3 – c1) + m3 (c1 – c2) = 0.
Show that the equation of the line passing through the origin and making an angle θ with the line `y = mx + c " is " y/c = (m+- tan theta)/(1 +- m tan theta)`.
Prove that the product of the lengths of the perpendiculars drawn from the points `(sqrt(a^2 - b^2), 0)` and `(-sqrta^2-b^2, 0)` to the line `x/a cos theta + y/b sin theta = 1` is `b^2`.
The hypotenuse of a right angled triangle has its ends at the points (1, 3) and (−4, 1). Find the equation of the legs (perpendicular sides) of the triangle that are parallel to the axes.
Find the equation of a line making an angle of 150° with the x-axis and cutting off an intercept 2 from y-axis.
Find the equation of the line which intercepts a length 2 on the positive direction of the x-axis and is inclined at an angle of 135° with the positive direction of y-axis.
Point R (h, k) divides a line segment between the axes in the ratio 1 : 2. Find the equation of the line.
Find the equation of a line for p = 8, α = 300°.
Find the equation of the line on which the length of the perpendicular segment from the origin to the line is 4 and the inclination of the perpendicular segment with the positive direction of x-axis is 30°.
Find the equation of the straight line on which the length of the perpendicular from the origin is 2 and the perpendicular makes an angle α with x-axis such that sin α = \[\frac{1}{3}\].
Find the equation of the straight line which makes a triangle of area \[96\sqrt{3}\] with the axes and perpendicular from the origin to it makes an angle of 30° with Y-axis.
Reduce the equation \[\sqrt{3}\] x + y + 2 = 0 to slope-intercept form and find slope and y-intercept;
Reduce the following equation to the normal form and find p and α in y − 2 = 0.
Put the equation \[\frac{x}{a} + \frac{y}{b} = 1\] to the slope intercept form and find its slope and y-intercept.
Show that the origin is equidistant from the lines 4x + 3y + 10 = 0; 5x − 12y + 26 = 0 and 7x + 24y = 50.
Find the point of intersection of the following pairs of lines:
bx + ay = ab and ax + by = ab.
Find the area of the triangle formed by the line y = m1 x + c1, y = m2 x + c2 and x = 0.
Find the area of the triangle formed by the line y = 0, x = 2 and x + 2y = 3.
Find the equations of the medians of a triangle, the equations of whose sides are:
3x + 2y + 6 = 0, 2x − 5y + 4 = 0 and x − 3y − 6 = 0
Prove that the following sets of three lines are concurrent:
3x − 5y − 11 = 0, 5x + 3y − 7 = 0 and x + 2y = 0
Find the conditions that the straight lines y = m1 x + c1, y = m2 x + c2 and y = m3 x + c3 may meet in a point.
Show that the straight lines L1 = (b + c) x + ay + 1 = 0, L2 = (c + a) x + by + 1 = 0 and L3 = (a + b) x + cy + 1 = 0 are concurrent.
Find the equation of a line which is perpendicular to the line \[\sqrt{3}x - y + 5 = 0\] and which cuts off an intercept of 4 units with the negative direction of y-axis.
If the image of the point (2, 1) with respect to the line mirror be (5, 2), find the equation of the mirror.
Find the projection of the point (1, 0) on the line joining the points (−1, 2) and (5, 4).
The equations of perpendicular bisectors of the sides AB and AC of a triangle ABC are x − y + 5 = 0 and x + 2y = 0 respectively. If the point A is (1, −2), find the equation of the line BC.
Determine whether the point (−3, 2) lies inside or outside the triangle whose sides are given by the equations x + y − 4 = 0, 3x − 7y + 8 = 0, 4x − y − 31 = 0 .
Find the equation of the line where length of the perpendicular segment from the origin to the line is 4 and the inclination of the perpendicular segment with the positive direction of x-axis is 30°.
The inclination of the line x – y + 3 = 0 with the positive direction of x-axis is ______.
If the line `x/"a" + y/"b"` = 1 passes through the points (2, –3) and (4, –5), then (a, b) is ______.
If the coordinates of the middle point of the portion of a line intercepted between the coordinate axes is (3, 2), then the equation of the line will be ______.
For specifying a straight line, how many geometrical parameters should be known?
A line passes through (2, 2) and is perpendicular to the line 3x + y = 3. Its y-intercept is ______.
Reduce the following equation into slope-intercept form and find their slopes and the y-intercepts.
y = 0
Reduce the following equation into intercept form and find their intercepts on the axes.
4x – 3y = 6
Reduce the following equation into normal form. Find their perpendicular distances from the origin and angle between perpendicular and the positive x-axis.
y − 2 = 0
Reduce the following equation into normal form. Find their perpendicular distances from the origin and angle between perpendicular and the positive x-axis.
x − y = 4
