Advertisements
Advertisements
प्रश्न
If p is the length of perpendicular from the origin on the line `x/a + y/b` = 1 and a2, p2, b2 are in A.P, then show that a4 + b4 = 0.
Advertisements
उत्तर
Given equation is `x/a + y/b` = 1
p = `|(0/a + 0/b - 1)/sqrt(1/a^2 + 1/b^2)|`
Squaring both sides, we have
p2 = `1/(1/a^2 + 1/b^2)`
⇒ `1/a^2 + 1/b^2 = 1/p^2` ......(i)
Since a2, p2, b2 are in A.P.
∴ 2p2 = a2 + b2
⇒ p2 = `(a^2 + b^2)/2`
⇒ `1/p^2 = 2/(a^2 + b^2)`
Putting the value of `1/p^2` is equation (i) we get,
`1/a^2 + 1/b^2 = 2/(a^2 + b^2)`
⇒ `(a^2 + b^2)/(a^2b^2) = 2/(a^2 + b^2)`
⇒ (a2 + b2)2 = 2a2b2
⇒ a4 + b4 + 2a2b2 = 2a2b2
⇒ a4 + b4 = 0.
Hence proved.
APPEARS IN
संबंधित प्रश्न
Find the value of x for which the points (x, –1), (2, 1) and (4, 5) are collinear.
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 value of p so that the three lines 3x + y – 2 = 0, px + 2y – 3 = 0 and 2x – y – 3 = 0 may intersect at one point.
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:
\[- \frac{\pi}{4}\]
Find the slope of the lines which make the following angle with the positive direction of x-axis:
\[\frac{2\pi}{3}\]
State whether the two lines in each of the following is parallel, perpendicular or neither.
Through (6, 3) and (1, 1); through (−2, 5) and (2, −5)
Find the slope of a line (i) which bisects the first quadrant angle (ii) which makes an angle of 30° with the positive direction of y-axis measured anticlockwise.
Using the method of slope, show that the following points are collinear A (4, 8), B (5, 12), C (9, 28).
Prove that the points (−4, −1), (−2, −4), (4, 0) and (2, 3) are the vertices of a rectangle.
If three points A (h, 0), P (a, b) and B (0, k) lie on a line, show that: \[\frac{a}{h} + \frac{b}{k} = 1\].
The slope of a line is double of the slope of another line. If tangents of the angle between them is \[\frac{1}{3}\],find the slopes of the other line.
Line through the points (−2, 6) and (4, 8) is perpendicular to the line through the points (8, 12) and (x, 24). Find the value of x.
Find the value of x for which the points (x, −1), (2, 1) and (4, 5) are collinear.
Find the equation of a straight line with slope − 1/3 and y-intercept − 4.
Find the equation of a straight line with slope −2 and intersecting the x-axis at a distance of 3 units to the left of origin.
Find the equation of the perpendicular to the line segment joining (4, 3) and (−1, 1) if it cuts off an intercept −3 from y-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.
Show that the perpendicular bisectors of the sides of a triangle are concurrent.
If the image of the point (2, 1) with respect to a line mirror is (5, 2), find the equation of the mirror.
Find the equation of the right bisector of the line segment joining the points (3, 4) and (−1, 2).
Find the tangent of the angle between the lines which have intercepts 3, 4 and 1, 8 on the axes respectively.
Show that the tangent of an angle between the lines \[\frac{x}{a} + \frac{y}{b} = 1 \text { and } \frac{x}{a} - \frac{y}{b} = 1\text { is } \frac{2ab}{a^2 - b^2}\].
Write the coordinates of the image of the point (3, 8) in the line x + 3y − 7 = 0.
The reflection of the point (4, −13) about the line 5x + y + 6 = 0 is
If m1 and m2 are slopes of lines represented by 6x2 - 5xy + y2 = 0, then (m1)3 + (m2)3 = ?
The line passing through (– 2, 0) and (1, 3) makes an angle of ______ with X-axis.
Find the equation of the straight line passing through (1, 2) and perpendicular to the line x + y + 7 = 0.
If the line joining two points A(2, 0) and B(3, 1) is rotated about A in anticlock wise direction through an angle of 15°. Find the equation of the line in new position.
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.
If one diagonal of a square is along the line 8x – 15y = 0 and one of its vertex is at (1, 2), then find the equation of sides of the square passing through this vertex.
Slope of a line which cuts off intercepts of equal lengths on the axes 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 (3, 4) and (2, – 6) are situated on the ______ of the line 3x – 4y – 8 = 0.
The points A(– 2, 1), B(0, 5), C(– 1, 2) are collinear.
The line `x/a + y/b` = 1 moves in such a way that `1/a^2 + 1/b^2 = 1/c^2`, where c is a constant. The locus of the foot of the perpendicular from the origin on the given line is x2 + y2 = c2.
| Column C1 | Column C2 |
| (a) The coordinates of the points P and Q on the line x + 5y = 13 which are at a distance of 2 units from the line 12x – 5y + 26 = 0 are |
(i) (3, 1), (–7, 11) |
| (b) The coordinates of the point on the line x + y = 4, which are at a unit distance from the line 4x + 3y – 10 = 0 are |
(ii) `(- 1/3, 11/3), (4/3, 7/3)` |
| (c) The coordinates of the point on the line joining A (–2, 5) and B (3, 1) such that AP = PQ = QB are |
(iii) `(1, 12/5), (-3, 16/5)` |
