Advertisements
Advertisements
प्रश्न
Show that the origin is equidistant from the lines 4x + 3y + 10 = 0; 5x − 12y + 26 = 0 and 7x + 24y = 50.
उत्तर
Let us write down the normal forms of the given lines.
First line: 4x + 3y + 10 = 0
\[\Rightarrow - 4x - 3y = 10\]
\[ \Rightarrow - \frac{4}{\sqrt{\left( - 4 \right)^2 + \left( - 3 \right)^2}}x - \frac{3}{\sqrt{\left( - 4 \right)^2 + \left( - 3 \right)^2}}y = \frac{10}{\sqrt{\left( - 4 \right)^2 + \left( - 3 \right)^2}} \left[ \text { Dividing both sides by } \sqrt{\left( \text { coefficient of x } \right)^2 + \left( \text { coefficient of y } \right)^2} \right]\]
\[ \Rightarrow - \frac{4}{5}x - \frac{3}{5}y = 2 \]
\[ \therefore p = 2\]
Second line: 5x − 12y + 26 = 0
\[\Rightarrow - 5x + 12y = 26\]
\[ \Rightarrow - \frac{5}{\sqrt{\left( - 5 \right)^2 + {12}^2}}x + \frac{12}{\sqrt{\left( - 5 \right)^2 + {12}^2}}y = \frac{26}{\sqrt{\left( - 5 \right)^2 + {12}^2}} \left[\text { Dividing both sides by } \sqrt{\left( \text { coefficient of x } \right)^2 + \left( \text { coefficient of y } \right)^2} \right]\]
\[ \Rightarrow - \frac{5}{13}x + \frac{12}{13}y = 2\]
\[ \therefore p = 2\]
Third line: 7x + 24y = 50
\[\Rightarrow \frac{7}{\sqrt{7^2 + {24}^2}}x + \frac{24}{\sqrt{7^2 + {24}^2}}y = \frac{50}{\sqrt{7^2 + {24}^2}} \left[ \text { Dividing both sides by }\sqrt{\left( \text { coefficient of x } \right)^2 + \left( \text { coefficient of y } \right)^2} \right]\]
\[ \Rightarrow \frac{7}{25}x + \frac{24}{25}y = 2\]
\[ \therefore p = 2\]
Hence, the origin is equidistant from the given lines.
APPEARS IN
संबंधित प्रश्न
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`
Find equation of the line perpendicular to the line x – 7y + 5 = 0 and having x intercept 3.
Two lines passing through the point (2, 3) intersects each other at an angle of 60°. If slope of one line is 2, find equation of the other line.
In the triangle ABC with vertices A (2, 3), B (4, –1) and C (1, 2), find the equation and length of altitude from the vertex A.
If p is the length of perpendicular from the origin to the line whose intercepts on the axes are a and b, then show that `1/p^2 = 1/a^2 + 1/b^2`.
If three lines whose equations are y = m1x + c1, y = m2x + c2 and y = m3x + c3 are concurrent, then show that m1(c2 – c3) + m2 (c3 – c1) + m3 (c1 – c2) = 0.
Find the equation of the line passing through the point of intersection of the lines 4x + 7y – 3 = 0 and 2x – 3y + 1 = 0 that has equal intercepts on the axes.
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 which is equidistant from the lines x = − 2 and x = 6.
Find the lines through the point (0, 2) making angles \[\frac{\pi}{3} \text { and } \frac{2\pi}{3}\] with the x-axis. Also, find the lines parallel to them cutting the y-axis at a distance of 2 units below the origin.
Find the equation of the line which intercepts a length 2 on the positive direction of the x-axis and is inclined at an angle of 135° with the positive direction of y-axis.
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 = 4, α = 150°.
Find the equation of a straight line on which the perpendicular from the origin makes an angle of 30° with x-axis and which forms a triangle of area \[50/\sqrt{3}\] with the axes.
Reduce the equation \[\sqrt{3}\] x + y + 2 = 0 to the normal form and find p and α.
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 \[x + y + \sqrt{2} = 0\].
Reduce the equation 3x − 2y + 6 = 0 to the intercept form and find the x and y intercepts.
Find the point of intersection of the following pairs of lines:
2x − y + 3 = 0 and x + y − 5 = 0
Find the point of intersection of the following pairs of lines:
\[y = m_1 x + \frac{a}{m_1} \text { and }y = m_2 x + \frac{a}{m_2} .\]
Show that the area of the triangle formed by the lines y = m1 x, y = m2 x and y = c is equal to \[\frac{c^2}{4}\left( \sqrt{33} + \sqrt{11} \right),\] where m1, m2 are the roots of the equation \[x^2 + \left( \sqrt{3} + 2 \right)x + \sqrt{3} - 1 = 0 .\]
Find the orthocentre of the triangle the equations of whose sides are x + y = 1, 2x + 3y = 6 and 4x − y + 4 = 0.
Prove that the following sets of three lines are concurrent:
15x − 18y + 1 = 0, 12x + 10y − 3 = 0 and 6x + 66y − 11 = 0
Prove that the following sets of three lines are concurrent:
3x − 5y − 11 = 0, 5x + 3y − 7 = 0 and x + 2y = 0
For what value of λ are the three lines 2x − 5y + 3 = 0, 5x − 9y + λ = 0 and x − 2y + 1 = 0 concurrent?
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.
If a, b, c are in A.P., prove that the straight lines ax + 2y + 1 = 0, bx + 3y + 1 = 0 and cx + 4y + 1 = 0 are concurrent.
Find the equation of the straight line which has y-intercept equal to \[\frac{4}{3}\] and is perpendicular to 3x − 4y + 11 = 0.
Find the coordinates of the foot of the perpendicular from the point (−1, 3) to the line 3x − 4y − 16 = 0.
Write the coordinates of the orthocentre of the triangle formed by the lines x2 − y2 = 0 and x + 6y = 18.
The equations of the sides AB, BC and CA of ∆ ABC are y − x = 2, x + 2y = 1 and 3x + y + 5 = 0 respectively. The equation of the altitude through B is
A (6, 3), B (−3, 5), C (4, −2) and D (x, 3x) are four points. If ∆ DBC : ∆ ABC = 1 : 2, then x is equal to
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 ______.
Find the equation of the lines which passes through the point (3, 4) and cuts off intercepts from the coordinate axes such that their sum is 14.
If the line `x/"a" + y/"b"` = 1 passes through the points (2, –3) and (4, –5), then (a, b) is ______.
The line which cuts off equal intercept from the axes and pass through the point (1, –2) is ______.
Reduce the following equation into slope-intercept form and find their slopes and the y-intercepts.
y = 0
