Home
Class 12
MATHS
If the coordinates of any two points Q1 ...

If the coordinates of any two points `Q_1 and Q_2` are `(x_1,y_1)` and `(x_2,y_2)`, respectively, then prove that `OQ_1xxOQ_2cos(angleQ_1OQ_2)=x_1x-2+y-1y_2`, whose O is the origin.

Text Solution

AI Generated Solution

To prove the equation \( OQ_1 \cdot OQ_2 \cdot \cos(\angle Q_1 O Q_2) = x_1 x_2 + y_1 y_2 \), where \( O \) is the origin and the coordinates of points \( Q_1 \) and \( Q_2 \) are \( (x_1, y_1) \) and \( (x_2, y_2) \) respectively, we will use the cosine rule in triangle \( OQ_1Q_2 \). ### Step-by-Step Solution: 1. **Identify the Points and Distances**: - Let \( O \) be the origin \( (0, 0) \). - The coordinates of point \( Q_1 \) are \( (x_1, y_1) \). - The coordinates of point \( Q_2 \) are \( (x_2, y_2) \). ...
Doubtnut Promotions Banner Mobile Dark
|

Topper's Solved these Questions

  • COORDINATE SYSYEM

    CENGAGE ENGLISH|Exercise Illustration1.14|1 Videos

  • COORDINATE SYSYEM

    CENGAGE ENGLISH|Exercise Illustration1.15|1 Videos

  • COORDINATE SYSYEM

    CENGAGE ENGLISH|Exercise Illustration1.12|1 Videos

  • COORDINATE SYSTEM

    CENGAGE ENGLISH|Exercise Multiple Correct Answers Type|2 Videos

  • CROSS PRODUCTS

    CENGAGE ENGLISH|Exercise DPP 2.2|13 Videos

Similar Questions

Explore conceptually related problems

If O is the origin and if the coordinates of any two points Q_1 \ a n d \ Q_2 are (x_1,y_1) \ a n d \ (x_2,y_2), respectively, prove that O Q_1.O Q_2cos/_Q_1O Q_2=x_1x_2+y_1y_2 .

If the line segment joining the points P(x_1, y_1)a n d\ Q(x_2, y_2) subtends an angle alpha at the origin O, prove that : O PdotO Q\ cosalpha=x_1x_2+y_1y_2dot

If [[x-y,2x-x_1],[2x-y,3x+y_1]], =[[-1,5],[0,13]] and coordinats of points P and Q be (x,y) and (x_1,y_1) respectively then P,Q=?

The coordinates of the ends of a focal chord of the parabola y^2=4a x are (x_1, y_1) and (x_2, y_2) . Then find the value of x_1x_2+y_1y_2 .

If the tangents and normals at the extremities of a focal chord of a parabola intersect at (x_1,y_1) and (x_2,y_2), respectively, then (a) x_1=y_2 (b) x_1=y_1 (c) y_1=y_2 (d) x_2=y_1

If O be the origin and if P(x_1, y_1) and P_2 (x_2, y_2) are two points, the OP_1 (OP_2) sin angle P_1OP_2 , is equal to

The coordinates of two points Pa n dQ are (x_1,y_1)a n d(x_2,y_2)a n dO is the origin. If the circles are described on O Pa n dO Q as diameters, then the length of their common chord is (a) (|x_1y_2+x_2y_1|)/(P Q) (b) (|x_1y_2-x_2y_1|)/(P Q) (|x_1x_2+y_1y_2|)/(P Q) (d) (|x_1x_2-y_1y_2|)/(P Q)

The coordinates of two points Pa n dQ are (x_1,y_1)a n d(x_2,y_2)a n dO is the origin. If the circles are described on O Pa n dO Q as diameters, then the length of their common chord is (|x_1y_2+x_2y_1|)/(P Q) (b) (|x_1y_2-x_2y_1|)/(P Q) (|x_1x_2+y_1y_2|)/(P Q) (d) (|x_1x_2-y_1y_2|)/(P Q)

If theta\ is the angle which the straight line joining the points (x_1, y_1)a n d\ (x_2, y_2) subtends at the origin, prove that tantheta=(x_2y_1-x_1y_2)/(x_1x_2+y_1y_2)\ a n dcostheta=(x_1x_2+y_1y_2)/(sqrt(x1 2 x2 2+y2 2x1 2+y1 2x2 2+ y1 2y2 2))

Three points A(x_1 , y_1), B (x_2, y_2) and C(x, y) are collinear. Prove that: (x-x_1) (y_2 - y_1) = (x_2 - x_1) (y-y_1) .

Home
Profile