RD Chapter 23 The Straight Lines Ex 23.8 Solutions
Question - 1 : - A line passes through a point A (1, 2) and makes an angle of 600 with the x–axis and intercepts the line x + y = 6 at the point P. Find AP.
Answer - 1 : -
Given:
(x1, y1)= A (1, 2), θ = 60°
Let us find thedistance AP.
By using the formula,
The equation of theline is given by:
Here, r represents thedistance of any point on the line from point A (1, 2).
The coordinate of anypoint P on this line are (1 + r/2, 2 + √3r/2)
It is clear that, Plies on the line x + y = 6
So,
∴ The value of AP is3(√3 – 1)
Question - 2 : - If the straight line through the point P(3, 4) makes an angle π/6 with the x–axis and meets the line 12x + 5y + 10 = 0 at Q, find the length PQ.
Answer - 2 : -
Given:
(x1, y1)= A (3, 4), θ = π/6 = 30°
Let us find the lengthPQ.
By using the formula,
The equation of theline is given by:
x – √3 y + 4√3 – 3 = 0
Let PQ = r
Then, the coordinateof Q are given by
Question - 3 : - A straight line drawn through the point A (2, 1) making anangle π/4 with positive x–axis intersects another line x + 2y + 1 = 0in the point B. Find length AB.
Answer - 3 : -
Given:
(x1, y1)= A (2, 1), θ = π/4 = 45°
Let us find the lengthAB.
By using the formula,
The equation of theline is given by:
x – y – 1 = 0
Let AB = r
Then, the coordinateof B is given by
Question - 4 : - A line a drawn through A (4, – 1) parallel to the line 3x – 4y + 1 = 0.Find the coordinates of the two points on this line which are at a distance of5 units from A.
Answer - 4 : -
Given:
(x1, y1)= A (4, -1)
Let us findCoordinates of the two points on this line which are at a distance of 5 unitsfrom A.
Given: Line 3x – 4y +1 = 0
4y = 3x + 1
y = 3x/4 + 1/4
Slope tan θ = 3/4
So,
Sin θ = 3/5
Cos θ = 4/5
The equation of theline passing through A (4, −1) and having slope ¾ is
By using the formula,
The equation of theline is given by:
3x – 4y = 16
Here, AP = r = ± 5
Thus, the coordinatesof P are given by
x
= ±4 + 4 and y = ±3 –1
x = 8, 0 and y = 2, –4
∴ The coordinates ofthe two points at a distance of 5 units from A are (8, 2) and (0, −4).
Question - 5 : - The straight line through P(x1, y1) inclined at an angle θ with the x–axis meets the line ax + by + c = 0 in Q. Find the length of PQ.
Answer - 5 : -
Given:
The equation of theline that passes through P(x1, y1) and makes anangle of θ with the x–axis.
Let us find the lengthof PQ.
By using the formula,
The equation of theline is given by:
