Advertisements
Advertisements
प्रश्न
In a triangle OAB,\[\angle\]AOB = 90º. If P and Q are points of trisection of AB, prove that \[{OP}^2 + {OQ}^2 = \frac{5}{9} {AB}^2\]
Advertisements
उत्तर
In triangle OAB,\[\angle\]AOB = 90º. P and Q are points of trisection of AB.
Taking O as the origin, let the position vectors of A and B be \[\vec{a}\] and \[\vec{b}\] respectively.
Since P and Q are the points of trisection of AB, so AP : PB = 1 : 2 and AQ : QB = 2 : 1.
Position vector of P, \[\vec{OP} = \frac{2 \vec{a} + \vec{b}}{3}\] ..................(Using section formula)
Position vector of Q,
\[\vec{OQ} = \frac{\vec{a} + 2 \vec{b}}{3}\]
\[\therefore \vec{a} . \vec{b} = 0\] ................(1)
Now,
\[{OP}^2 + {OQ}^2 \]
\[ = \left| \vec{OP} \right|^2 + \left| \vec{OQ} \right|^2 \]
\[ = \left( \frac{2 \vec{a} + \vec{b}}{3} \right) . \left( \frac{2 \vec{a} + \vec{b}}{3} \right) + \left( \frac{\vec{a} + 2 \vec{b}}{3} \right) . \left( \frac{\vec{a} + 2 \vec{b}}{3} \right)\]
\[ = \frac{4 \left| \vec{a} \right|^2 + 4 \vec{a} . \vec{b} + \left| \vec{b} \right|^2 + \left| \vec{a} \right|^2 + 4 \vec{a} . \vec{b} + 4 \left| \vec{b} \right|^2}{9}\]
\[= \frac{5 \left| \vec{a} \right|^2 + 5 \left| \vec{b} \right|^2}{9}.............. \left[ \text{ Using } \left( 1 \right) \right]\]
\[ = \frac{5}{9}\left( \left| \vec{a} \right|^2 + \left| \vec{b} \right|^2 \right)\]
\[ = \frac{5}{9} \left| \vec{AB} \right|^2 ........................\left[ \text{ Using Pythagoras Theorem } \right]\]
\[ = \frac{5}{9} {AB}^2\]
APPEARS IN
संबंधित प्रश्न
Show that the points A (1, –2, –8), B (5, 0, –2) and C (11, 3, 7) are collinear, and find the ratio in which B divides AC.
If the origin is the centroid of the triangle whose vertices are A(2, p, –3), B(q, –2, 5) and C(–5, 1, r), then find the values of p, q, r.
(Pythagoras's Theorem) Prove by vector method that in a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Prove by vector method that the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of its sides.
Prove using vectors: The quadrilateral obtained by joining mid-points of adjacent sides of a rectangle is a rhombus.
If the median to the base of a triangle is perpendicular to the base, then triangle is isosceles.
Let `A(bara)` and `B(barb)` are any two points in the space and `R(barr)` be a point on the line segment AB dividing it internally in the ratio m : n, then prove that `barr = (mbarb + nbara)/(m + n)`.
Find the position vector of point R which divides the line joining the points P and Q whose position vectors are `2hati - hatj + 3hatk` and `- 5hati + 2hatj - 5hatk` in the ratio 3:2 is internally.
Find the position vector of point R which divides the line joining the points P and Q whose position vectors are `2hat"i" - hat"j" + 3hat"k"` and `- 5hat"i" + 2hat"j" - 5hat"k"` in the ratio 3 : 2 is externally.
If the points A(3, 0, p), B(–1, q, 3) and C(–3, 3, 0) are collinear, then find
- the ratio in which the point C divides the line segment AB
- the values of p and q.
If the centroid of a tetrahedron OABC is (1, 2, - 1) where A(a, 2, 3), B(1, b, 2), C(2, 1, c), find the distance of P(a, b, c) from origin.
Prove that `(bar"a" xx bar"b").(bar"c" xx bar"d")` =
`|bar"a".bar"c" bar"b".bar"c"|`
`|bar"a".bar"d" bar"b".bar"d"|.`
If `bara, barb` and `barc` are position vectors of the points A, B, C respectively and `5bara - 3barb - 2barc = bar0`, then find the ratio in which the point C divides the line segement BA.
If G(a, 2, −1) is the centroid of the triangle with vertices P(1, 2, 3), Q(3, b, −4) and R(5, 1, c) then find the values of a, b and c
Prove that altitudes of a triangle are concurrent
If the plane 2x + 3y + 5z = 1 intersects the co-ordinate axes at the points A, B, C, then the centroid of Δ ABC is ______.
If G(3, -5, r) is centroid of triangle ABC where A(7, - 8, 1), B(p, q, 5) and C(q + 1, 5p, 0) are vertices of a triangle then values of p, q, rare respectively.
P is the point of intersection of the diagonals of the parallelogram ABCD. If O is any point, then `overline"OA" + overline"OB" + overline"OC" + overline"OD"` = ______
If G and G' are the centroids of the triangles ABC and A'B'C', then `overline("A""A"^') + overline("B""B"^') + overline("C""C"^')` is equal to ______
The image of the point (1, 6, 3) in the line `x/1 = (y - 1)/2 = (z - 2)/3` is ______
If G`(overlineg)` is the centroid, `H(overlineh)` is the orthocentre and P`(overlinep)` is the circumcentre of a triangle and `xoverlinep + yoverlineh + zoverlineg = 0`, then ______
If A, B, C are the vertices of a triangle whose position vectors are `overline("a"),overline("b"),overline("c")` and G is the centroid of the `triangle ABC,` then `overline("GA")+overline("GB")+overline("GC")` is ______.
Find the unit vector in the diret:tion of the vector `veca = hati + hatj + 2hatk`
If D, E, F are the mid points of the sides BC, CA and AB respectively of a triangle ABC and 'O' is any point, then, `|vec(AD) + vec(BE) + vec(CF)|`, is ______.
In ΔABC the mid-point of the sides AB, BC and CA are respectively (l, 0, 0), (0, m, 0) and (0, 0, n). Then, `("AB"^2 + "BC"^2 + "CA"^2)/("l"^2 + "m"^2 + "n"^2)` is equal to ______.
If `overlinea, overlineb, overlinec` are the position vectors of the points A, B, C respectively and `5overlinea + 3overlineb - 8overlinec = overline0` then find the ratio in which the point C divides the line segment AB.
The position vector of points A and B are `6 bar "a" + 2 bar "b" and bar "a" - 3 bar"b"`. If the point C divided AB in the ratio 3 : 2, show that the position vector of C is `3 bar "a" - bar "b".`
Find the ratio in which the point C divides segment AB, if `5bara + 4barb - 9barc = bar0`
Let `A(bara)` and `B(barb)` be any two points in the space and `R(barr)` be the third point on the line AB dividing the segment AB externally in the ratio m : n, then prove that `barr = (mbarb - nbara)/(m - n)`.
The position vector of points A and B are `6bara + 2 barb and bara - 3 barb`. If point C divides AB in the ratio 3 : 2, then show that the position vector of C is `3bara - barb`.
