Advertisements
Advertisements
Question
Find the equation of the lines through the point (3, 2) which make an angle of 45° with the line x –2y = 3.
Advertisements
Solution
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
RELATED QUESTIONS
Find a point on the x-axis, which is equidistant from the points (7, 6) and (3, 4).
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).
Find the value of x for which the points (x, –1), (2, 1) and (4, 5) are collinear.
Find the angle between the x-axis and the line joining the points (3, –1) and (4, –2).
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 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:
\[\frac{2\pi}{3}\]
Find the slope of a line passing through the following point:
(−3, 2) and (1, 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 are parallel, perpendicular or neither.
Through (9, 5) and (−1, 1); through (3, −5) and (8, −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.
Using the method of slope, show that the following points are collinear A (4, 8), B (5, 12), C (9, 28).
What can be said regarding a line if its slope is zero ?
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: \[\frac{a}{h} + \frac{b}{k} = 1\].
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 the perpendicular to the line segment joining (4, 3) and (−1, 1) if it cuts off an intercept −3 from y-axis.
Find the equation of the strainght line intersecting y-axis at a distance of 2 units above the origin and making an angle of 30° with the positive direction of the x-axis.
If the image of the point (2, 1) with respect to a line mirror is (5, 2), find the equation of the mirror.
Find the image of the point (3, 8) with respect to the line x + 3y = 7 assuming the line to be a plane mirror.
Find the angles between the following pair of straight lines:
3x − y + 5 = 0 and x − 3y + 1 = 0
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.
Show that the line a2x + ay + 1 = 0 is perpendicular to the line x − ay = 1 for all non-zero real values of a.
Write the coordinates of the image of the point (3, 8) in the line x + 3y − 7 = 0.
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 reflection of the point (4, −13) about the line 5x + y + 6 = 0 is
If x + y = k is normal to y2 = 12x, then k is ______.
Point of the curve y2 = 3(x – 2) at which the normal is parallel to the line 2y + 4x + 5 = 0 is ______.
Show that the tangent of an angle between the lines `x/a + y/b` = 1 and `x/a - y/b` = 1 is `(2ab)/(a^2 - b^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.
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.
One vertex of the equilateral triangle with centroid at the origin and one side as x + y – 2 = 0 is ______.
Equations of the lines through the point (3, 2) and making an angle of 45° with the line x – 2y = 3 are ______.
The vertex of an equilateral triangle is (2, 3) and the equation of the opposite side is x + y = 2. Then the other two sides are y – 3 = `(2 +- sqrt(3)) (x - 2)`.
The lines whose vector equations are `r = 2hati - 3hatj + 7hatk + lambda (2hati + phatj + 5hatk) and r = hati - 2hatj + 3hatk + µ(3hati + phatj + phatk)` are perpendicular for all values of λ and µ if p =
