Advertisements
Advertisements
प्रश्न
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.
विकल्प
True
False
Advertisements
उत्तर
This statement is False.
Explanation:
Equation of any line perpendicular to x sec θ + y cosec θ = a is
x cosec θ – y sec θ = k .......(i)
If equation (i) passes through (a cos3θ, a sin3θ) then
a cos3θ.cosec θ – a sin3θ.secθ = k
⇒ `(a cos^3 theta)/sintheta - (asin^3theta)/costheta` = k
∴ Required equation is
x cos θ – y sin θ = `(a cos^3 theta)/sintheta - (asin^3theta)/costheta`
⇒ `x/sintheta - y/costheta = a[(cos^4theta - sin^4theta)/(sintheta costheta)]`
⇒ `(xcostheta - ysintheta)/(sintheta costheta) = a[((cos^2theta + sin^2theta)(cos^2theta - sin^2theta))/(sintheta costheta)]`
⇒ x cos θ – y sin θ = a(cos2θ – sin2θ)
⇒ x cos θ – y sin θ = a cos 2θ.
APPEARS IN
संबंधित प्रश्न
Find the equation of a line equidistant from the lines y = 10 and y = − 2.
Find the equation of the straight line passing through the point (6, 2) and having slope − 3.
Find the equation of the straight line passing through (−2, 3) and inclined at an angle of 45° with the x-axis.
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 :
(0, −a) and (b, 0)
Find the equation of the straight lines passing through the following pair of point :
(a, b) and (a + b, a − b)
Find the equations of the sides of the triangles the coordinates of whose angular point is respectively (0, 1), (2, 0) and (−1, −2).
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 length L (in centimeters) of a copper rod is a linear function of its celsius temperature C. In an experiment, if L = 124.942 when C = 20 and L = 125.134 when C = 110, express L in terms of C.
Find the equation to the straight line cutting off intercepts 3 and 2 from the axes.
Find the equation to the straight line cutting off intercepts − 5 and 6 from the axes.
Find the equation of the straight line which passes through (1, −2) and cuts off equal intercepts on 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.
A straight line passes through the point (α, β) and this point bisects the portion of the line intercepted between the axes. Show that the equation of the straight line is \[\frac{x}{2 \alpha} + \frac{y}{2 \beta} = 1\].
Find the equation of the line which passes through the point (3, 4) and is such that the portion of it intercepted between the axes is divided by the point in the ratio 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.
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.
A line is such that its segment between the straight lines 5x − y − 4 = 0 and 3x + 4y − 4 = 0 is bisected at the point (1, 5). Obtain its equation.
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 the straight line through the point (α, β) and perpendicular to the line lx + my + n = 0.
Find the equation of the straight line perpendicular to 5x − 2y = 8 and which passes through the mid-point of the line segment joining (2, 3) and (4, 5).
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 distance of the point (1, 2) from the straight line with slope 5 and passing through the point of intersection of x + 2y = 5 and x − 3y = 7.
Find the equations to the sides of an isosceles right angled triangle the equation of whose hypotenues is 3x + 4y = 4 and the opposite vertex is the point (2, 2).
Find the equation of the straight line drawn through the point of intersection of the lines x + y = 4 and 2x − 3y = 1 and perpendicular to the line cutting off intercepts 5, 6 on the axes.
Prove that the family of lines represented by x (1 + λ) + y (2 − λ) + 5 = 0, λ being arbitrary, pass through a fixed point. Also, find the fixed point.
Find the equation of the straight line which passes through the point of intersection of the lines 3x − y = 5 and x + 3y = 1 and makes equal and positive intercepts on the axes.
If the diagonals of the quadrilateral formed by the lines l1x + m1y + n1 = 0, l2x + m2y + n2 = 0, l1x + m1y + n1' = 0 and l2x + m2y + n2' = 0 are perpendicular, then write the value of l12 − l22 + m12 − m22.
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 straight line which passes through the point (−4, 3) such that the portion of the line between the axes is divided internally by the point in the ratio 5 : 3 is
The equation of the line passing through (1, 5) and perpendicular to the line 3x − 5y + 7 = 0 is
In what direction should a line be drawn through the point (1, 2) so that its point of intersection with the line x + y = 4 is at a distance `sqrt(6)/3` from the given point.
The equation of the line passing through the point (1, 2) and perpendicular to the line x + y + 1 = 0 is ______.
