A straight line passes through the points A (-5, 2) and B (3, -6). It intersects the coordinate axes at points C and D as shown in the given diagram. M is a point on AB which divides AB in the ratio 1 : 2 and D is on the Y-axis
Find :
(i) the co-ordinates of points C and D.
(ii) the co-ordinates of point M.
Answers
Answer:
Slope (m) of a non-vertical line passing through the points (x1 ,y1 )(x1 , y1 ) and (x2 , y2)(x2 , y2) is given by is given by m =y1−y2x1−x2= m =y1−y2x1−x2= x1≠x2x1≠x2.
If a line makes an angle á with the positive direction of x-axis, then the slope of the line is given by m =tanα, α≠90om =tanα, α≠90o
Slope of horizontal line is zero and slope of vertical line is undefined.
An acute angle (say θ) between lines L1 and L2L1 and L2with slopes m1 and m2m1 and m2 is given by tanθ=∣∣m2−m11+m1m2∣∣tanθ=|m2−m11+m1m2|, 1+m1m2≠01+m1m2≠0
Two lines are parallel if and only if their slopes are equal i.e., m1=m2m1=m2
Two lines are perpendicularif and only if product of their slopes is –1, i.e., mm2=−1m1.m2=−1
Three points A, B and C are collinear, if and only if slope of AB = slope of BC.
Equation of the horizontal line having distance a from the x-axis is eithery = a or y = – a.
Equation of the vertical line having distance b from the y-axis is eitherx = b or x = – b.
The point (x, y) lies on the line with slope m and through the fixed point (xo,y0 ),(xo, y0 ), if and only if its coordinates satisfy the equation.
Various forms of equations of a line:
Two points form: Equation of the line passing through the points (x1,y1)(x1, y1) and ((x2, y2)(x2, y2) is given by y−y1=y2−y1x2−x1(x−x1)y−y1=y2−y1x2−x1(x−x1)
Slope-Intercept form: The point (x, y) on the line with slope m and y-intercept c lies on the line if and only if y=mx +cy=mx +c.
If a line with slope m makes x-intercept d. Then equation of the line is y=m(x -d)y=m(x -d).
Intercept form: Equation of a line making intercepts a and b on the x-and y-axis, respectively, is xa+yb=1xa+yb=1.
Normal form: The equation of the line having normal distance from origin p and angle between normal and the positive x−axis ωx−axis ωis given by x cosω +ysin ω=p x cosω +ysin ω=p
General Equation of a Line: Any equation of the form Ax + By + C = 0, with A and B are not zero, simultaneously, is called the general linear equation or general equation of a line.
Working Rule for reducing general form into the normal form:
(i) Shift constant ‘C’ to the R.H.S. and get Ax+By=−CAx+By=−C
(ii) If the R.H.S. is not positive, then make it positive by multiplying the whole equation by -1.
(iii) Divide both sides of equation by A2+B2−−−−
Answer:
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