Advertisements
Advertisements
Question
Find the length of the perpendicular from the origin to the straight line joining the two points whose coordinates are (a cos α, a sin α) and (a cos β, a sin β).
Advertisements
Solution
Equation of the line passing through (acosα, asinα) and (acosβ, asinβ) is
\[y - asin\alpha = \frac{asin\beta - asin\alpha}{acos\beta - acos\alpha}\left( x - acos\alpha \right)\]
\[ \Rightarrow y - asin\alpha = \frac{sin\beta - sin\alpha}{cos\beta - cos\alpha}\left( x - acos\alpha \right)\]
\[ \Rightarrow y - asin\alpha = \frac{2\cos\left( \frac{\beta + \alpha}{2} \right)\sin\left( \frac{\beta - \alpha}{2} \right)}{2\sin\left( \frac{\beta + \alpha}{2} \right)\sin\left( \frac{\alpha - \beta}{2} \right)}\left( x - acos\alpha \right)\]
\[ \Rightarrow y - asin\alpha = - \cot\left( \frac{\beta + \alpha}{2} \right)\left( x - acos\alpha \right)\]
\[ \Rightarrow y - asin\alpha = - \cot\left( \frac{\alpha + \beta}{2} \right)\left( x - acos\alpha \right)\]
\[\Rightarrow x\cot\left( \frac{\alpha + \beta}{2} \right) + y - asin\alpha - acos\alpha \cot\left( \frac{\alpha + \beta}{2} \right) = 0\]
The distance of the line from the origin is
\[d = \left| \frac{- asin\alpha - acos\alpha \cot\left( \frac{\alpha + \beta}{2} \right)}{\sqrt{\cot^2 \left( \frac{\alpha + \beta}{2} \right) + 1}} \right|\]
\[ \Rightarrow d = \left| \frac{asin\alpha + acos\alpha \cot\left( \frac{\alpha + \beta}{2} \right)}{\sqrt{{cosec}^2 \left( \frac{\alpha + \beta}{2} \right)}} \right| \left( \because {cosec}^2 \theta = 1 + \cot^2 \theta \right)\]
\[\Rightarrow d = a\left| \sin\left( \frac{\alpha + \beta}{2} \right)sin\alpha + cos\alpha \cos\left( \frac{\alpha + \beta}{2} \right) \right| \]
\[ \Rightarrow d = a\left| sin\alpha \sin\left( \frac{\alpha + \beta}{2} \right) + cos\alpha \cos\left( \frac{\alpha + \beta}{2} \right) \right| \]
\[ \Rightarrow d = a\left| \cos\left( \frac{\alpha + \beta}{2} - \alpha \right) \right| = a\cos\left( \frac{\beta - \alpha}{2} \right) = a\cos\left( \frac{\alpha - \beta}{2} \right)\]
Hence, the required distance is \[a\cos\left( \frac{\alpha - \beta}{2} \right)\]
APPEARS IN
RELATED QUESTIONS
Find the equation of the line parallel to x-axis and passing through (3, −5).
Find the equation of the straight line which passes through the point (1,2) and makes such an angle with the positive direction of x-axis whose sine is \[\frac{3}{5}\].
Prove that the perpendicular drawn from the point (4, 1) on the join of (2, −1) and (6, 5) divides it in the ratio 5 : 8.
Find the equation of the straight lines passing through the following pair of point:
(a, b) and (a + c sin α, b + c cos α)
Find the equation of the straight lines passing through the following pair of point :
(a, b) and (a + b, a − b)
Find the equation of the straight lines passing through the following pair of point :
(at1, a/t1) and (at2, a/t2)
Find the equation to the straight line which bisects the distance between the points (a, b), (a', b') and also bisects the distance between the points (−a, b) and (a', −b').
In what ratio is the line joining the points (2, 3) and (4, −5) divided by the line passing through the points (6, 8) and (−3, −2).
The owner of a milk store finds that he can sell 980 litres milk each week at Rs 14 per liter and 1220 liters of milk each week at Rs 16 per liter. Assuming a linear relationship between selling price and demand, how many liters could he sell weekly at Rs 17 per liter.
Find the equation to the straight line cutting off intercepts − 5 and 6 from the axes.
Find the equation to the straight line which passes through the point (5, 6) and has intercepts on the axes
(i) equal in magnitude and both positive,
(ii) equal in magnitude but opposite in sign.
Find the equation to the straight line which cuts off equal positive intercepts on the axes and their product is 25.
Find the equation of the line, which passes through P (1, −7) and meets the axes at A and Brespectively so that 4 AP − 3 BP = 0.
Find the equation of the line passing through the point (2, 2) and cutting off intercepts on the axes whose sum is 9.
Find the equation of the straight line which passes through the point P (2, 6) and cuts the coordinate axes at the point A and B respectively so that \[\frac{AP}{BP} = \frac{2}{3}\] .
Find the equations of the straight lines each of which passes through the point (3, 2) and cuts off intercepts a and b respectively on X and Y-axes such that a − b = 2.
Find the equation of the straight line passing through the origin and bisecting the portion of the line ax + by + c = 0 intercepted between the coordinate axes.
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.
Three sides AB, BC and CA of a triangle ABC are 5x − 3y + 2 = 0, x − 3y − 2 = 0 and x + y − 6 = 0 respectively. Find the equation of the altitude through the vertex A.
Find the equation of a line passing through (3, −2) and perpendicular to the line x − 3y + 5 = 0.
The line 2x + 3y = 12 meets the x-axis at A and y-axis at B. The line through (5, 5) perpendicular to AB meets the x-axis and the line AB at C and E respectively. If O is the origin of coordinates, find the area of figure OCEB.
Find the equations to the straight lines which pass through the origin and are inclined at an angle of 75° to the straight line \[x + y + \sqrt{3}\left( y - x \right) = a\].
Find the equations of the straight lines passing through (2, −1) and making an angle of 45° with the line 6x + 5y − 8 = 0.
Write the area of the triangle formed by the coordinate axes and the line (sec θ − tan θ) x + (sec θ + tan θ) y = 2.
Find the locus of the mid-points of the portion of the line x sinθ+ y cosθ = p intercepted between the axes.
The equation of the line passing through (1, 5) and perpendicular to the line 3x − 5y + 7 = 0 is
The inclination of the straight line passing through the point (−3, 6) and the mid-point of the line joining the point (4, −5) and (−2, 9) is
Find the equation of the line passing through the point of intersection of 2x + y = 5 and x + 3y + 8 = 0 and parallel to the line 3x + 4y = 7.
A straight line moves so that the sum of the reciprocals of its intercepts made on axes is constant. Show that the line passes through a fixed point.
Find the equations of the lines through the point of intersection of the lines x – y + 1 = 0 and 2x – 3y + 5 = 0 and whose distance from the point (3, 2) is `7/5`
The equation of the line passing through the point (1, 2) and perpendicular to the line x + y + 1 = 0 is ______.
Equation of the line passing through the point (a cos3θ, a sin3θ) and perpendicular to the line x sec θ + y cosec θ = a is x cos θ – y sin θ = a sin 2θ.
The straight line 5x + 4y = 0 passes through the point of intersection of the straight lines x + 2y – 10 = 0 and 2x + y + 5 = 0.
The lines ax + 2y + 1 = 0, bx + 3y + 1 = 0 and cx + 4y + 1 = 0 are concurrent if a, b, c are in G.P.
