Advertisements
Advertisements
प्रश्न
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.
उत्तर
Let AM be perpendicular to line BC.
(i) Slope of line BC
= `("y"_2 - "y"_1)/("x"_2 - "x"_1)`
= `(2 + 1)/(1 - 4)`
= `3/ (-3)`
= −1
AM ⊥ BC,
∴ Slope of perpendicular AM = `(-1)/"m"`
= `(-1)/(-1)`
= 1
Line AM passes through point A and slope = 1.
∴ equation of AM
y – y1 = m(x – x1)
y – 3 = 1(x – 2)
or x – y + 1 = 0
(ii) Equation of line BC passing through points B(4, –1) and C(1, 2)
`"y"- "y"_1 = ("y"_2 - "y"_1)/("x"_2 - "x"_1)("x" - "x"_1)`
y + 1 = `(2 + 1)/(1 - 4) ("x" - 4)`
= `3/(-3) ("x" - 4)`
= −x + 4
x + y − 3 = 0
∴ Length of perpendicular AM from point A to BC
= `(2 + 3 -3)/sqrt(1^2 + 1^2)` ..........`[∵ "d" = ("ax"_1 + "by"_1 + "c")/sqrt("a"^2 + "b"^2)]`
= `2/sqrt2`
= `sqrt2`
APPEARS IN
संबंधित प्रश्न
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.
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?
The hypotenuse of a right angled triangle has its ends at the points (1, 3) and (−4, 1). Find the equation of the legs (perpendicular sides) of the triangle that are parallel to the axes.
Find the equation of a line which is equidistant from the lines x = − 2 and x = 6.
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 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°.
Find the value of θ and p, if the equation x cos θ + y sin θ = p is the normal form of the line \[\sqrt{3}x + y + 2 = 0\].
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.
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 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.
Find the values of θ and p, if the equation x cos θ + y sin θ = p is the normal form of the line \[\sqrt{3}x + y + 2 = 0\].
Find the area of the triangle formed by the line y = 0, x = 2 and x + 2y = 3.
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 \[x^2 + \left( \sqrt{3} + 2 \right)x + \sqrt{3} - 1 = 0 .\]
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.
Find the equation of the right bisector of the line segment joining the points (a, b) and (a1, b1).
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).
Write the coordinates of the orthocentre of the triangle formed by the lines xy = 0 and x + y = 1.
Write the area of the figure formed by the lines a |x| + b |y| + c = 0.
The figure formed by the lines ax ± by ± c = 0 is
Two vertices of a triangle are (−2, −1) and (3, 2) and third vertex lies on the line x + y = 5. If the area of the triangle is 4 square units, then the third vertex is
The centroid of a triangle is (2, 7) and two of its vertices are (4, 8) and (−2, 6). The third vertex is
Find the equation of a line which passes through the point (2, 3) and makes an angle of 30° with the positive direction of x-axis.
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°.
If the intercept of a line between the coordinate axes is divided by the point (–5, 4) in the ratio 1 : 2, then find the equation of the line.
Find the equation of the line which passes through the point (– 4, 3) and the portion of the line intercepted between the axes is divided internally in the ratio 5 : 3 by this point.
For specifying a straight line, how many geometrical parameters should be known?
A line passes through (2, 2) and is perpendicular to the line 3x + y = 3. Its y-intercept 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 normal form. Find their perpendicular distances from the origin and angle between perpendicular and the positive x-axis.
x − y = 4
