Advertisements
Advertisements
प्रश्न
Find the coordinates of the foot of the perpendicular from the point (−1, 3) to the line 3x − 4y − 16 = 0.
उत्तर
Let A (−1, 3) be the given point.
Also, let M (h, k) be the foot of the perpendicular drawn from A (−1, 3) to the line 3x − 4y − 16 = 0
Point M (h, k) lies on the line 3x − 4y − 16 = 0
3h − 4k − 16 = 0 ... (1)
Lines 3x − 4y − 16 = 0 and AM are perpendicular.
\[\therefore\] \[\frac{k - 3}{h + 1} \times \frac{3}{4} = - 1\]
\[\Rightarrow 4h + 3k - 5 = 0\] ... (2)
Solving eq (1) and eq (2) by cross multiplication, we get:
\[\frac{h}{20 + 48} = \frac{k}{- 64 + 15} = \frac{1}{9 + 16}\]
\[ \Rightarrow a = \frac{68}{25}, b = - \frac{49}{25}\]
Hence, the coordinates of the foot of perpendicular are \[\left( \frac{68}{25}, - \frac{49}{25} \right)\].
APPEARS IN
संबंधित प्रश्न
Reduce the following equation into intercept form and find their intercepts on the axes.
3y + 2 = 0
Find equation of the line perpendicular to the line x – 7y + 5 = 0 and having x intercept 3.
Find angles between the lines `sqrt3x + y = 1 and x + sqrt3y = 1`.
Find the coordinates of the foot of perpendicular from the point (–1, 3) to the line 3x – 4y – 16 = 0.
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 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 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.
In what ratio, the line joining (–1, 1) and (5, 7) is divided by the line x + y = 4?
Find the equation of the right bisector of the line segment joining the points (3, 4) and (−1, 2).
Find the equation of a line for p = 8, α = 300°.
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.
Reduce the following equation to the normal form and find p and α in \[x - y + 2\sqrt{2} = 0\].
Put the equation \[\frac{x}{a} + \frac{y}{b} = 1\] to the slope intercept form and find its slope and y-intercept.
Reduce the lines 3 x − 4 y + 4 = 0 and 2 x + 4 y − 5 = 0 to the normal form and hence find which line is nearer to the origin.
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:
bx + ay = ab and ax + by = ab.
Find the area of the triangle formed by the line y = 0, x = 2 and x + 2y = 3.
Prove that the following sets of three lines are concurrent:
3x − 5y − 11 = 0, 5x + 3y − 7 = 0 and x + 2y = 0
If the lines p1 x + q1 y = 1, p2 x + q2 y = 1 and p3 x + q3 y = 1 be concurrent, show that the points (p1, q1), (p2, q2) and (p3, q3) are collinear.
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 perpendicular bisector of the line joining the points (1, 3) and (3, 1).
Find the equation of a line which is perpendicular to the line \[\sqrt{3}x - y + 5 = 0\] and which cuts off an intercept of 4 units with the negative direction of y-axis.
If the image of the point (2, 1) with respect to the line mirror be (5, 2), find the equation of the mirror.
Find the projection of the point (1, 0) on the line joining the points (−1, 2) and (5, 4).
Find the values of α so that the point P (α2, α) lies inside or on the triangle formed by the lines x − 5y+ 6 = 0, x − 3y + 2 = 0 and x − 2y − 3 = 0.
Write the coordinates of the orthocentre of the triangle formed by the lines x2 − y2 = 0 and x + 6y = 18.
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
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 passes through (2, 2) and is perpendicular to the line 3x + y = 3. Its y-intercept 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.
x + 7y = 0
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.
3x + 2y – 12 = 0
