Advertisements
Advertisements
प्रश्न
The two lines ax + by = c and a′x + b′y = c′ are perpendicular if ______.
पर्याय
aa′ + bb′ = 0
ab′ = ba′
ab + a′b′ = 0
ab′ + ba′ = 0
Advertisements
उत्तर
The two lines ax + by = c and a′x + b′y = c′ are perpendicular if aa′ + bb′ = 0
Explanation:
Slope of the line ax + by = c is `(-a)/b`.
And the slope of the line a′x + b′y = c′ is `(-a"'")/(b"'")`.
The lines are perpendicular if tan θ = `3/(5 - x)`
`(-a)/b (-a"'")/(b"'")` = − 1 or aa' bb′ + = 0 (Why?)
APPEARS IN
संबंधित प्रश्न
Find the value of x for which the points (x, –1), (2, 1) and (4, 5) are collinear.
Without using distance formula, show that points (–2, –1), (4, 0), (3, 3) and (–3, 2) are vertices of a parallelogram.
The slope of a line is double of the slope of another line. If tangent of the angle between them is `1/3`, find the slopes of the lines.
Find the equation of a line drawn perpendicular to the line `x/4 + y/6 = 1`through the point, where it meets the y-axis.
Find the equation of the lines through the point (3, 2) which make an angle of 45° with the line x –2y = 3.
Find the slope of the lines which make the following angle with the positive direction of x-axis:
\[\frac{3\pi}{4}\]
Find the slope of a line passing through the following point:
\[(a t_1^2 , 2 a t_1 ) \text { and } (a t_2^2 , 2 a t_2 )\]
State whether the two lines in each of the following are parallel, perpendicular or neither.
Through (9, 5) and (−1, 1); through (3, −5) and (8, −3)
What can be said regarding a line if its slope is positive ?
Show that the line joining (2, −5) and (−2, 5) is perpendicular to the line joining (6, 3) and (1, 1).
Without using Pythagoras theorem, show that the points A (0, 4), B (1, 2) and C (3, 3) are the vertices of a right angled triangle.
Without using the distance formula, show that points (−2, −1), (4, 0), (3, 3) and (−3, 2) are the vertices of a parallelogram.
Find the angle between the X-axis and the line joining the points (3, −1) and (4, −2).
By using the concept of slope, show that the points (−2, −1), (4, 0), (3, 3) and (−3, 2) are the vertices of a parallelogram.
Find the equation of the strainght line intersecting y-axis at a distance of 2 units above the origin and making an angle of 30° with the positive direction of the x-axis.
Find the equations of the straight lines which cut off an intercept 5 from the y-axis and are equally inclined to the axes.
Find the equation of the right bisector of the line segment joining the points (3, 4) and (−1, 2).
Find the image of the point (3, 8) with respect to the line x + 3y = 7 assuming the line to be a plane mirror.
Find the angles between the following pair of straight lines:
3x + 4y − 7 = 0 and 4x − 3y + 5 = 0
If θ is the angle which the straight line joining the points (x1, y1) and (x2, y2) subtends at the origin, prove that \[\tan \theta = \frac{x_2 y_1 - x_1 y_2}{x_1 x_2 + y_1 y_2}\text { and } \cos \theta = \frac{x_1 x_2 + y_1 y_2}{\sqrt{{x_1}^2 + {y_1}^2}\sqrt{{x_2}^2 + {y_2}^2}}\].
Prove that the straight lines (a + b) x + (a − b ) y = 2ab, (a − b) x + (a + b) y = 2ab and x + y = 0 form an isosceles triangle whose vertical angle is 2 tan−1 \[\left( \frac{a}{b} \right)\].
Write the coordinates of the image of the point (3, 8) in the line x + 3y − 7 = 0.
The acute angle between the medians drawn from the acute angles of a right angled isosceles triangle is
If the slopes of the lines given by the equation ax2 + 2hxy + by2 = 0 are in the ratio 5 : 3, then the ratio h2 : ab = ______.
If the slope of a line passing through the point A(3, 2) is `3/4`, then find points on the line which are 5 units away from the point A.
Find the equation to the straight line passing through the point of intersection of the lines 5x – 6y – 1 = 0 and 3x + 2y + 5 = 0 and perpendicular to the line 3x – 5y + 11 = 0.
If the equation of the base of an equilateral triangle is x + y = 2 and the vertex is (2, – 1), then find the length of the side of the triangle.
A variable line passes through a fixed point P. The algebraic sum of the perpendiculars drawn from the points (2, 0), (0, 2) and (1, 1) on the line is zero. Find the coordinates of the point P.
P1, P2 are points on either of the two lines `- sqrt(3) |x|` = 2 at a distance of 5 units from their point of intersection. Find the coordinates of the foot of perpendiculars drawn from P1, P2 on the bisector of the angle between the given lines.
The coordinates of the foot of perpendiculars from the point (2, 3) on the line y = 3x + 4 is given by ______.
One vertex of the equilateral triangle with centroid at the origin and one side as x + y – 2 = 0 is ______.
Equations of the lines through the point (3, 2) and making an angle of 45° with the line x – 2y = 3 are ______.
The points A(– 2, 1), B(0, 5), C(– 1, 2) are collinear.
The equation of the line through the intersection of the lines 2x – 3y = 0 and 4x – 5y = 2 and
| Column C1 | Column C2 |
| (a) Through the point (2, 1) is | (i) 2x – y = 4 |
| (b) Perpendicular to the line (ii) x + y – 5 = 0 x + 2y + 1 = 0 is |
(ii) x + y – 5 = 0 |
| (c) Parallel to the line (iii) x – y –1 = 0 3x – 4y + 5 = 0 is |
(iii) x – y –1 = 0 |
| (d) Equally inclined to the axes is | (iv) 3x – 4y – 1 = 0 |
A ray of light coming from the point (1, 2) is reflected at a point A on the x-axis and then passes through the point (5, 3). The co-ordinates of the point A is ______.
