Advertisements
Advertisements
प्रश्न
Find the equation of the lines through the point (3, 2) which make an angle of 45° with the line x –2y = 3.
Advertisements
उत्तर
Let the equation of line AB be: x – 2y = 3
or y = `1/2 "x" - 3`
Then the slope of line AB = `1/2`
Let the PA line make an angle of 45° with it and its slope = m.
`± tan 45= ("m" - 1/2)/(1 + "m" xx 1/2)`
or `± 1 = (2"m" - 1)/("m" + 2)`
Taking +ve sign, 1 = `(2"m" - 1)/("m" + 2)`
or 2m – 1 = m + 2
∴ m = 3
2m –1
Taking – ve sign, –1 = `(2"m" - 1)/("m" + 2)`
or 2m – 1 = –m – 2
∴ 3m = –1
or m = `(-1)/3`
Hence, the equation of line PA is where point P = (3, 2) and m = `(-1)/3`.
y – 2 = `- (-1)/3 ("x" - 3)`
3y – 6 = – x + 3
or x + 3y – 9 = 0
Now when the slope is m = 3, then the equation of the line from the point P(3, 2),
y – 2 = 3(x – 3)
y – 2 = 3x – 9
or 3x – y – 7 = 0
APPEARS IN
संबंधित प्रश्न
Find the slope of a line, which passes through the origin, and the mid-point of the line segment joining the points P (0, –4) and B (8, 0).
Without using the Pythagoras theorem, show that the points (4, 4), (3, 5) and (–1, –1) are the vertices of a right angled triangle.
Without using distance formula, show that points (–2, –1), (4, 0), (3, 3) and (–3, 2) are vertices of a parallelogram.
If three point (h, 0), (a, b) and (0, k) lie on a line, show that `q/h + b/k = 1`
Find the values of k for which the line (k–3) x – (4 – k2) y + k2 –7k + 6 = 0 is
- Parallel to the x-axis,
- Parallel to the y-axis,
- Passing through the origin.
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)
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).
Show that the line joining (2, −5) and (−2, 5) is perpendicular to the line joining (6, 3) and (1, 1).
Prove that the points (−4, −1), (−2, −4), (4, 0) and (2, 3) are the vertices of a rectangle.
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 − 1/3 and y-intercept − 4.
Find the equations of the bisectors of the angles between the coordinate axes.
Find the coordinates of the orthocentre of the triangle whose vertices are (−1, 3), (2, −1) and (0, 0).
Show that the perpendicular bisectors of the sides of a triangle are concurrent.
If the image of the point (2, 1) with respect to a line mirror is (5, 2), find the equation of the mirror.
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 angles between the following pair of straight lines:
3x − y + 5 = 0 and x − 3y + 1 = 0
The acute angle between the medians drawn from the acute angles of a right angled isosceles triangle 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 = ______.
If x + y = k is normal to y2 = 12x, then k is ______.
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). Find the coordinates of the point A.
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 ______.
The coordinates of the foot of the perpendicular from the point (2, 3) on the line x + y – 11 = 0 are ______.
If p is the length of perpendicular from the origin on the line `x/a + y/b` = 1 and a2, p2, b2 are in A.P, then show that a4 + b4 = 0.
The tangent of angle between the lines whose intercepts on the axes are a, – b and b, – a, respectively, is ______.
If the vertices of a triangle have integral coordinates, then the triangle can not be equilateral.
| 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 |
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 ______.
The three straight lines ax + by = c, bx + cy = a and cx + ay = b are collinear, if ______.
