Advertisements
Advertisements
Question
The reflection of the point (4, −13) about the line 5x + y + 6 = 0 is
Options
(−1, −14)
(3, 4)
(0, 0)
(1, 2)
Advertisements
Solution
Let the reflection point be A(h, k)
Now, the mid point of line joining (h, k) and (4, −13) will lie on the line 5x + y + 6 = 0
\[\therefore 5\left( \frac{h + 4}{2} \right) + \frac{k - 13}{2} + 6 = 0\]
\[ \Rightarrow 5h + 20 + k - 13 + 12 = 0\]
\[ \Rightarrow 5h + k + 19 = 0 . . . . . \left( 1 \right)\]
Now, the slope of the line joining points (h, k) and (4,−13) are perpendicular to the line 5x + y + 6 = 0.
slope of the line = −5
slope of line joining by points (h, k) and (4,−13)
\[\frac{k + 13}{h - 4}\]
\[\therefore \frac{k + 13}{h - 4}\left( - 5 \right) = - 1\]
\[ \Rightarrow 5k - h + 69 = 0 . . . . . \left( 2 \right)\]
Solving (1) and (2), we get
h = −1 and k = −14
Hence, the correct answer is option (a).
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.
Without using the Pythagoras theorem, show that the points (4, 4), (3, 5) and (–1, –1) are the vertices of a right angled triangle.
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.
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 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:
\[- \frac{\pi}{4}\]
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 is parallel, perpendicular or neither.
Through (3, 15) and (16, 6); through (−5, 3) and (8, 2).
What can be said regarding a line if its slope is negative?
Prove that the points (−4, −1), (−2, −4), (4, 0) and (2, 3) are the vertices of a rectangle.
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 value of x for which the points (x, −1), (2, 1) and (4, 5) are collinear.
Find the angle between X-axis and the line joining the points (3, −1) and (4, −2).
A quadrilateral has vertices (4, 1), (1, 7), (−6, 0) and (−1, −9). Show that the mid-points of the sides of this quadrilateral form a parallelogram.
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 altitudes of a ∆ ABC whose vertices are A (1, 4), B (−3, 2) and C (−5, −3).
Find the image of the point (3, 8) with respect to the line x + 3y = 7 assuming the line to be a plane mirror.
If θ is the angle which the straight line joining the points (x1, y1) and (x2, y2) subtends at the origin, prove that \[\tan \theta = \frac{x_2 y_1 - x_1 y_2}{x_1 x_2 + y_1 y_2}\text { and } \cos \theta = \frac{x_1 x_2 + y_1 y_2}{\sqrt{{x_1}^2 + {y_1}^2}\sqrt{{x_2}^2 + {y_2}^2}}\].
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.
Write the coordinates of the image of the point (3, 8) in the line x + 3y − 7 = 0.
The angle between the lines 2x − y + 3 = 0 and x + 2y + 3 = 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 m1 and m2 are slopes of lines represented by 6x2 - 5xy + y2 = 0, then (m1)3 + (m2)3 = ?
Point of the curve y2 = 3(x – 2) at which the normal is parallel to the line 2y + 4x + 5 = 0 is ______.
The two lines ax + by = c and a′x + b′y = c′ are perpendicular if ______.
The reflection of the point (4, – 13) about the line 5x + y + 6 = 0 is ______.
Find the angle between the lines y = `(2 - sqrt(3)) (x + 5)` and y = `(2 + sqrt(3))(x - 7)`
Slope of a line which cuts off intercepts of equal lengths on the axes is ______.
The points (3, 4) and (2, – 6) are situated on the ______ of the line 3x – 4y – 8 = 0.
If the vertices of a triangle have integral coordinates, then the triangle can not be equilateral.
Line joining the points (3, – 4) and (– 2, 6) is perpendicular to the line joining the points (–3, 6) and (9, –18).
| 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)` |
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 |
The three straight lines ax + by = c, bx + cy = a and cx + ay = b are collinear, if ______.
