Advertisements
Advertisements
प्रश्न
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.
उत्तर
Let the equation of line AB be, y = mx + c
Slope of line AB = m
From O, perpendicular OC is drawn on line AB, which meets at point C(−1, 2).
∴ Slope of perpendicular line OC = `-1/"m"`
Now the equation of line OC,
y – 0 = `-1/"m"("x" - 0)`
or x + my = 0
Slope of OC = `(2 - 0)/(-1 -1) = -2`
Slope of perpendicular line OC = `-1/"m"`
The point C (−1, 2) lies on the following line:
y = mx + c
⇒ 2 = –m + c
Putting m = `1/2`,
2 = `- 1/2 + "c"`
∴ C = `2 + 1/2`
= `5/2`
Hence, m = `1/2`, C = `5/2`.
APPEARS IN
संबंधित प्रश्न
Find equation of the line parallel to the line 3x – 4y + 2 = 0 and passing through the point (–2, 3).
Find angles between the lines `sqrt3x + y = 1 and x + sqrt3y = 1`.
Prove that the line through the point (x1, y1) and parallel to the line Ax + By + C = 0 is A (x –x1) + B (y – y1) = 0.
Find the equation of the right bisector of the line segment joining the points (3, 4) and (–1, 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.
Show that the equation of the line passing through the origin and making an angle θ with the line `y = mx + c " is " y/c = (m+- tan theta)/(1 +- m tan theta)`.
A person standing at the junction (crossing) of two straight paths represented by the equations 2x – 3y+ 4 = 0 and 3x + 4y – 5 = 0 wants to reach the path whose equation is 6x – 7y + 8 = 0 in the least time. Find equation of the path that he should follow.
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 side BC of the triangle ABC whose vertices are (−1, −2), (0, 1) and (2, 0) respectively. Also, find the equation of the median through (−1, −2).
Point R (h, k) divides a line segment between the axes in the ratio 1 : 2. Find the equation of the line.
Find the equation of a line for p = 4, α = 150°.
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 \[\frac{5}{12}\].
If the straight line through the point P (3, 4) makes an angle π/6 with the x-axis and meets the line 12x + 5y + 10 = 0 at Q, find the length PQ.
Reduce the equation \[\sqrt{3}\] x + y + 2 = 0 to slope-intercept form and find slope and y-intercept;
Reduce the equation\[\sqrt{3}\] x + y + 2 = 0 to intercept form and find intercept on the axes.
Find the values of θ and p, if the equation x cos θ + y sin θ = p is the normal form of the line \[\sqrt{3}x + y + 2 = 0\].
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:
\[y = m_1 x + \frac{a}{m_1} \text { and }y = m_2 x + \frac{a}{m_2} .\]
Find the area of the triangle formed by the line y = 0, x = 2 and x + 2y = 3.
Prove that the lines \[y = \sqrt{3}x + 1, y = 4 \text { and } y = - \sqrt{3}x + 2\] form an equilateral triangle.
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.
For what value of λ are the three lines 2x − 5y + 3 = 0, 5x − 9y + λ = 0 and x − 2y + 1 = 0 concurrent?
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 right bisector of the line segment joining the points (a, b) and (a1, b1).
Find the projection of the point (1, 0) on the line joining the points (−1, 2) and (5, 4).
Find the values of the parameter a so that the point (a, 2) is an interior point of the triangle formed by the lines x + y − 4 = 0, 3x − 7y − 8 = 0 and 4x − y − 31 = 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
The number of real values of λ for which the lines x − 2y + 3 = 0, λx + 3y + 1 = 0 and 4x − λy + 2 = 0 are concurrent 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
The figure formed by the lines ax ± by ± c = 0 is
A point equidistant from the line 4x + 3y + 10 = 0, 5x − 12y + 26 = 0 and 7x+ 24y − 50 = 0 is
Find the equation of the line where length of the perpendicular segment from the origin to the line is 4 and the inclination of the perpendicular segment with the positive direction of x-axis is 30°.
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.
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 ______.
The line which cuts off equal intercept from the axes and pass through the point (1, –2) is ______.
Locus of the mid-points of the portion of the line x sin θ + y cos θ = p intercepted between the axes is ______.
