Advertisements
Advertisements
प्रश्न
Find the equation of straight line passing through (−2, −7) and having an intercept of length 3 between the straight lines 4x + 3y = 12 and 4x + 3y = 3.
उत्तर
Here,
\[\left( x_1 , y_1 \right) = A\left( - 2, - 7 \right)\]
So, the equation of the line is
\[\frac{x - x_1}{cos\theta} = \frac{y - y_1}{sin\theta}\]
\[ \Rightarrow \frac{x + 2}{cos\theta} = \frac{y + 7}{sin\theta}\]
Let the required line intersect the lines 4x + 3y = 3 and 4x + 3y = 12 at P1 and P2.
Let AP1 = r1 and AP2 = r2
Then, the coordinates of P1 and P2 are given by
\[\frac{x - x_1}{cos\theta} = \frac{y - y_1}{sin\theta}={r_1}\] and \[ \Rightarrow \frac{x + 2}{cos\theta} = \frac{y + 7}{sin\theta}= {r_2}\], respectively.
Thus, the coordinates of P1 and P2 are \[\left( - 2 + r_1 cos\theta, - 7 + r_1 sin\theta \right) \text { and } \left( - 2 + r_2 cos\theta, - 7 + r_2 sin\theta \right)\], respectively.
Clearly, the points P1 and P2 lie on the lines 4x + 3y = 3 and 4x + 3y = 12
\[4\left( - 2 + r_1 cos\theta \right) + 3\left( - 7 + r_1 sin\theta \right) = 3 and 4\left( - 2 + r_2 cos\theta \right) + 3\left( - 7 + r_2 sin\theta \right) = 12\]
\[ \Rightarrow r_1 = \frac{32}{4cos\theta + 3sin\theta} \text { and } r_2 = \frac{41}{4cos\theta + 3sin\theta}\]
\[\text { Here }, A P_2 - A P_1 = 3 \Rightarrow r_2 - r_1 = 3\]
\[ \Rightarrow \frac{41}{4cos\theta + 3sin\theta} - \frac{32}{4cos\theta + 3sin\theta} = 3\]
\[ \Rightarrow 3 = 4cos\theta + 3sin\theta\]
\[ \Rightarrow 3\left( 1 - sin\theta \right) = 4cos\theta\]
\[ \Rightarrow 9\left( 1 + \sin^2 \theta - 2sin\theta \right) = 16 \cos^2 \theta = 16\left( 1 - \sin^2 \theta \right)\]
\[ \Rightarrow 25 \sin^2 \theta - 18sin\theta - 7 = 0\]
\[ \Rightarrow \left( sin\theta - 1 \right)\left( 25sin\theta + 7 \right) = 0\]
\[ \Rightarrow sin\theta = 1, sin\theta = - \frac{7}{25}\]
\[ \Rightarrow cos\theta = 0, cos\theta = \frac{24}{25}\]
Thus, the equation of the required line is
\[x + 2 = 0\text { or } \frac{x + 2}{\frac{24}{25}} = \frac{y + 7}{\frac{- 7}{25}}\]
\[ \Rightarrow x + 2 = 0 \text { or } 7x + 24y + 182 = 0\]
APPEARS IN
संबंधित प्रश्न
Find the equations of the straight lines which pass through (4, 3) and are respectively parallel and perpendicular to the x-axis.
Find the equation of the line passing through (0, 0) with slope m.
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 \[\frac{3}{5}\].
Find the equations to the altitudes of the triangle whose angular points are A (2, −2), B (1, 1) and C (−1, 0).
Find the equation of the line passing through the point (−3, 5) and perpendicular to the line joining (2, 5) and (−3, 6).
Find the equation of the straight lines passing through the following pair of point :
(at1, a/t1) and (at2, a/t2)
Find the equations of the sides of the triangles the coordinates of whose angular point is respectively (0, 1), (2, 0) and (−1, −2).
Find the equations of the medians of a triangle, the coordinates of whose vertices are (−1, 6), (−3, −9) and (5, −8).
By using the concept of equation of a line, prove that the three points (−2, −2), (8, 2) and (3, 0) are collinear.
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 equations to the straight lines which go through the origin and trisect the portion of the straight line 3 x + y = 12 which is intercepted between the axes of coordinates.
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 straight line which passes through the point (−3, 8) and cuts off positive intercepts on the coordinate axes whose sum is 7.
Find the equation of a line which passes through the point (22, −6) and is such that the intercept of x-axis exceeds the intercept of y-axis by 5.
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.
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 passing through the point of intersection of the lines 5x − 6y − 1 = 0 and 3x + 2y + 5 = 0 and perpendicular to the line 3x − 5y + 11 = 0 .
Find the equation of the straight lines passing through the origin and making an angle of 45° with the straight line \[\sqrt{3}x + y = 11\].
Find the equations to the straight lines passing through the point (2, 3) and inclined at and angle of 45° to the line 3x + y − 5 = 0.
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 equations of the two straight lines through (1, 2) forming two sides of a square of which 4x+ 7y = 12 is one diagonal.
Two sides of an isosceles triangle are given by the equations 7x − y + 3 = 0 and x + y − 3 = 0 and its third side passes through the point (1, −10). Determine the equation of the third side.
Show that the straight lines given by (2 + k) x + (1 + k) y = 5 + 7k for different values of k pass through a fixed point. Also, find that point.
Find the equation of the straight line passing through the point of intersection of 2x + y − 1 = 0 and x + 3y − 2 = 0 and making with the coordinate axes a triangle of area \[\frac{3}{8}\] sq. units.
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 a, b, c are in A.P., then the line ax + by + c = 0 passes through a fixed point. Write the coordinates of that point.
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 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.
If a, b, c are in A.P., then the straight lines ax + by + c = 0 will always pass through ______.
The equation of the line joining the point (3, 5) to the point of intersection of the lines 4x + y – 1 = 0 and 7x – 3y – 35 = 0 is equidistant from the points (0, 0) and (8, 34).
