Advertisements
Advertisements
प्रश्न
Prove that every straight line has an equation of the form Ax + By + C = 0, where A, B and C are constants.
उत्तर
Given a straight line
Either it cuts the y-axis, or is parallel to or coincident with it.
We know that the equation of a line which cuts the y-axis (i.e., it has y-intercept) can be put in the form y = mx + b; further
If the line is parallel to or coincident with the y-axis
Its equation is of the form x = x1
Where x = 0 in the case of coincidence.
Both of these equations are of the form given in the problem and hence the proof.
APPEARS IN
संबंधित प्रश्न
Reduce the following equation into intercept form and find their intercepts on the axes.
3y + 2 = 0
Reduce the following equation into normal form. Find their perpendicular distances from the origin and angle between perpendicular and the positive x-axis.
`x – sqrt3y + 8 = 0`
The perpendicular from the origin to the line y = mx + c meets it at the point (–1, 2). Find the values of m and c.
If p and q are the lengths of perpendiculars from the origin to the lines x cos θ – y sin θ = k cos 2θ and xsec θ+ y cosec θ = k, respectively, prove that p2 + 4q2 = k2.
In what ratio, the line joining (–1, 1) and (5, 7) is divided by the line x + y = 4?
Prove that the product of the lengths of the perpendiculars drawn from the points `(sqrt(a^2 - b^2), 0)` and `(-sqrta^2-b^2, 0)` to the line `x/a cos theta + y/b sin theta = 1` is `b^2`.
Find the equation of a line making an angle of 150° with the x-axis and cutting off an intercept 2 from y-axis.
Find the equation of a line that has y-intercept −4 and is parallel to the line joining (2, −5) and (1, 2).
Find the equation of the bisector of angle A of the triangle whose vertices are A (4, 3), B (0, 0) and C(2, 3).
Find the equation of a line for p = 8, α = 300°.
Find the equation of the straight line upon which the length of the perpendicular from the origin is 2 and the slope of this perpendicular is \[\frac{5}{12}\].
Find the equation of the straight line which makes a triangle of area \[96\sqrt{3}\] with the axes and perpendicular from the origin to it makes an angle of 30° with Y-axis.
Reduce the following equation to the normal form and find p and α in \[x + \sqrt{3}y - 4 = 0\] .
Reduce the following equation to the normal form and find p and α in x − 3 = 0.
Put the equation \[\frac{x}{a} + \frac{y}{b} = 1\] to the slope intercept form and find its slope and y-intercept.
Reduce the equation 3x − 2y + 6 = 0 to the intercept form and find the x and y intercepts.
Find the area of the triangle formed by the line x + y − 6 = 0, x − 3y − 2 = 0 and 5x − 3y + 2 = 0.
Find the coordinates of the incentre and centroid of the triangle whose sides have the equations 3x− 4y = 0, 12y + 5x = 0 and y − 15 = 0.
Prove that the following sets of three lines are concurrent:
15x − 18y + 1 = 0, 12x + 10y − 3 = 0 and 6x + 66y − 11 = 0
If the three lines ax + a2y + 1 = 0, bx + b2y + 1 = 0 and cx + c2y + 1 = 0 are concurrent, show that at least two of three constants a, b, c are equal.
Find the coordinates of the foot of the perpendicular from the point (−1, 3) to the line 3x − 4y − 16 = 0.
If the lines ax + 12y + 1 = 0, bx + 13y + 1 = 0 and cx + 14y + 1 = 0 are concurrent, then a, b, c are in
A point equidistant from the line 4x + 3y + 10 = 0, 5x − 12y + 26 = 0 and 7x+ 24y − 50 = 0 is
A line passes through P(1, 2) such that its intercept between the axes is bisected at P. The equation of the line is ______.
For what values of a and b the intercepts cut off on the coordinate axes by the line ax + by + 8 = 0 are equal in length but opposite in signs to those cut off by the line 2x – 3y + 6 = 0 on the axes.
A line cutting off intercept – 3 from the y-axis and the tangent at angle to the x-axis is `3/5`, its equation is ______.
Reduce the following equation into slope-intercept form and find their slopes and the y-intercepts.
6x + 3y – 5 = 0
Reduce the following equation into intercept form and find their intercepts on the axes.
4x – 3y = 6
