28278 + Interview Questions in Maths in Class 12 Page 115 InterviewSolution

5701.	Determine order and degree (if defined) of the following differential equations y'' + (y)^2 + 2y = 0
Answer» Solution :The highest order derivative in the DIFFERENTIAL equation is y.. and its DEGREE is 1. `therefore` The order and the degree of the differential equation are 2 and 1 respectively.

Discussion

5702.	(i) The base of an open rectangular box is square and its volume is256 cm^(3). Find the dimensions ofthis box ifwe want to use least material for construction: (ii) A window is in theform of a rectangle surmounted by a semi-circle. Its perimeter is 40m. Find the dimensions of this window from which maximum light canadmit.
Answer» ANSWER :(i) 8 cm, 8 cm, 4 cm (ii) Length = `80/(pi+4)" metre, Breadth "40/(pi+4)" metre,Radius "40/(pi+4)` metre

Discussion

5703.	The points A (2, 3), B(3, 5), C (7, 7) and D(4, 7) are such that
Answer» ABCD is a parallelogram A, B, C and D are COLLINEAR D lies inside the triangle ABC D lies on the boundary of the triangle ABC Answer :D

Discussion

5704.	If (x)/((1+x^(2))(3-2x))=(A)/(3-2x)+(Bx+C)/(1+x^(2)) then C=
Answer» `(-1)/(13)` `(2)/(13)` `(-2)/(13)` `(1)/(13)` ANSWER :C

Discussion

5705.	Write the set of points of continuity of f(x)=\|x-1\|+\|x+1\|
Answer» ANSWER :R

Discussion

5706.	A point P (x,y)moves in in xy plane in such a way that sqrt(2) le \|x + y\| + \|x - y\| le 3sqrt(2) . Area of region representing all possible position of point P is equal to:
Answer» Solution :`1LE\|(x+y)/(SQRT2)\|+\|(x-y)/(sqrt2)\|LE3` `1le\|X\|+\|Y\|le3""("Rotation through an angle of "(PI)/(4))` `"Area "=4((1)/(2)xx3xx-(1)/(2)xx1xx1)` `=4[(9)/(2)-(1)/(2)]` `=16"(unit)"^(2)`

Discussion

5707.	C_(1):x^(2)+y^(2)=r^(2)and C_(2):(x^(2))/(16)+(y^(2))/(9)=1 interset at four distinct points A,B,C, and D. Their common tangents form a peaallelogram A'B'C'D'. If A'B'C'D' is a square, then the ratio of the area of circle C_(1) to the area of circumcircle of DeltaA'B'C' is
Answer» `9//16` `3//4` `1//2` none of these Solution :If A'B'C'D' is a SQUARE, then TANGENTS are `y=+-x+-5` for which DIAGONAL length A'C' is 10. Then the area of circumericle of `DeltaA'B'C' "is"25pi` Also, the area of the circle `C_(1)` is `25//pi` . Hence the require ration is 1/2

Discussion

5708.	If A = (0, 0), B = (3, 0), C = (0, 4) are the vertices of a triangle then match the following {:("I. Centroid",(a)(1,1)),("II. Orthocentre",(b)(1,4//3)),("III. Circumcentre",(c)(0,0)),("IV. Incentre",(d)(4,5)),(,(e)(3//2,2)):}
Answer» a, B, C, d a, b, d, E b, c, e, a c, d, e, b ANSWER :C

Discussion

5709.	The value of underset(0)overset(pi//2)int (sin^(3)x)/(sin x + cos x)dxis (pi-1)/(k) . The value of k is ________.
Answer» Solution :`int_(0)^(2) (sin^(3)x)/(sin x+cosx)dx=I` (say) `RARR I= int_(0)^(pi//2) (COS^(3)x)/(sin x + cos x) dx` `rArr 2I = int_(0)^(pi//2) ( sin^(3) x+cos^(3)x)/(sin x+ cos x) dx rArr 2I = int_(0)^(pi//2) ( sin^(2)x+cos^(2)x-sin x cos x) dx` `rArr 2 I = int_(0)^(pi//2)(1-1/2 sin 2 x) dx rArr 2 I = x + (cos 2x)/(4) ]_(0)^(pi//2)` `rArr 2I=(pi/2 - 1/4) - (1/4)= pi/4-1/2 rArr I=(pi-1)/(4)`

Discussion

5710.	How many of them are divisible by 25
Answer» ANSWER :6

Discussion

5711.	At some integer points a polynomial with integer coefficients take values 1, 2 and 3. Prove that thereexist not more than one integer at which the polynomial is equal to 5.
Answer» ANSWER :21

Discussion

5712.	Let f : R^(+) rarr R satisfies the functional equation f(xy) = e^(xy - x - y) {e^(y) f(x) + e^(x) f(y)}, AA x, y in R^(+). If f'(1) = e, determine f(x).
Answer» ANSWER :`F(X) = E^(x) LOG \|x\|`

Discussion

5713.	Choose the correct answer int dx/(x(x^2+1) =
Answer» ` log\|x\|-1/2log(x^2+1)+C` `log\|x\|+1/2log(x^2+1)+c` `- log\|x\|+1/2log(x^2+1)+c` `1/2 log\|x\|+log(x^2+1)+c` ANSWER :A

Discussion

5714.	AL, BM and CN are perpendicular from angular points of a triangle ABC on the opposite sides BC, CA and AB respectively. Delta is the area of triangle ABC, (r ) and R are the inradius and circumradius. IF areas of Delta 's AMN, BNL and CLM are Delta_(1), Delta_(2)and Delta_(3) respectively, then the valur of Delta _(1)+Delta_(2)+Delta _(3) is
Answer» `DELTA (2+2 COS A cos B cos C)` `Delta(2+ 2 SIN A sin B sin C)` `Delta (1-2 cos A cos B cos C)` `Delta (1-2 sin A sin B sin C)` ANSWER :C

Discussion

5715.	intTan(sqrt((1-x)/(1+x)))dx=
Answer» ANSWER :`(1)/(2)[XCOS^(-1)x-sqrt(1-x^(2))]+C`

Discussion

5716.	AL, BM and CN are perpendicular from angular points of a triangle ABC on the opposite sides BC, CA and AB respectively. Delta is the area of triangle ABC, (r ) and R are the inradius and circumradius. Radius os the circum circle of Delta LMN is
Answer» 2R R `R/2` `R/4` ANSWER :B

Discussion

5717.	AL, BM and CN are perpendicular from angular points of a triangle ABC on the opposite sides BC, CA and AB respectively. Delta is the area of triangle ABC, (r ) and R are the inradius and circumradius. If area of Delta LMNis Delta ', then the value of (Delta')/(Delta) is
Answer» `2 SIN A sin B sin C` `2 COS A cos B cos C` `sin A sin B sin C` `cos A cos B cos C` Answer :D

Discussion

5718.	AL, BM and CN are perpendicular from angular points of a triangle ABC on the opposite sides BC, CA and AB respectively. Delta is the area of triangle ABC, (r ) and R are the inradius and circumradius. If radis of the incircle of Delta LMN is r', then the valur of r' sec A sec B sec C is
Answer» 4R 3R 2R R Answer :A

Discussion

5719.	AL, BM and CN are perpendicular from angular points of a triangle ABC on the opposite sides BC, CA and AB respectively. Delta is the area of triangle ABC, (r ) and R are the inradius and circumradius. If perimeters of DeltaLMN and Delta ABC an lamda and mu,then the value of (lamda)/(mu) is
Answer» `r/R` `R/r` `(RR)/(DELTA)` `(Delta)/(rR)` ANSWER :B

Discussion

5720.	For given events A and B if P(A) = 0.25 and P(B) = 0.50 probability that both event occurs together is 0.14. Then ........ is the probability of an event that A and B does not occurs.
Answer» 0.39 0.25 0.11 None of these Answer :A

Discussion

5721.	Each question given below has four possible answers, out of which only one is correct. Choose the correct one. (hati+hatk)xx(hati+hatj+hatk) = ___
Answer» `hati-hatk` `hatk-hati` `hatk-2hati-hatj` 2 Answer :B

Discussion

5722.	If the function f(x) & g(x) are defined on R to R such that f (x) = {{:( 0 , x in "rational") , (x "," , x in "irrational"):} g(x) = {{:( 0 "," , x in "irrational" ) , ( x "," , x in "rational" ):} then (f-g) (x) is
Answer» ONE-one and onto one-one but nor onto onto but not one- one NEITHER one-one nor onto Answer :A

Discussion

5723.	If f_(1)(x)=1-sqrt(1-x^(2)) and g_(1)'(x)=(sqrt2-1)sqrt(1-x^(2)) for 0le x le 1, then find the area bounded between the curves f_(2)(x)=max{f_(1)(t), 0le tlex}, g_(2)(x)=min{g_(1)(t), 0 le t le x)" for 0le x le 1 and y - axis.
Answer» ANSWER :`(pi-1)/(2SQRT2)" SQ. UNITS"`

Discussion

5724.	Equation of the plane that contains the lines vecr = (hati +hatj) + lambda(hati + 2hatj - hatk) and vecr = (hati + hatj) + mu(-hati + hatj-2hatk) is
Answer» `vecr(2hati + hatj - 3hatk)=-4` `vecr xx (-hati + hatj + HATK) =vec0` `vecr(-hati + hatj + hatk) =0` None of these Solution :The lines are parallel to the vectors `vecb_(1) = hati + 2hatj -hatk` and `vecb_(2) = -hati + hatj - 2HATK`. Therefore, the plane is NORMAL to the vector `vecn = vecb_(1) xx vecb_(2) = \|{:(hati, hatj,hatk),(1,2,-1),(-1,1,-2):}\|=-3hati + 3hatj + 3hatk` The required plane PASSES through `(hati + hatj)` and is normal to the vector `vecn`. Therefore, its equation is `vecr.vecn = veca. vecn` `rArr vecr.(-3hati + 3hatj + 3hatk) = (hati + hatj).(-3hati + 3hatj + 3hatk)` `rArr vecr.(-3hati + 3hatj + 3hatk) = -3+3` `rArr vecr.(-hati + hatj +hatk)=0`

Discussion

5725.	If sum_(r=0)^(n-1)log_(2)((r+2)/(r+1)) prod_(r=10)^(99)log_(r) (r+1), then 'n' is equal to
Answer» 4 3 5 6

Discussion

5726.	If f(x)=2x^(3)+mx^(2)-13x+nare 2, 3 roots of theequation f(x)=0, then the value of m and n are
Answer» `-5, -30` `-5, 30` `5, 30` NONE of these ANSWER :B

Discussion

5727.	If f is defined in [1,3] by f(x)=x^(3)+bx^(2)+ax such that f(1)-f(3)=0 and f'(c)=0, "where" c=2+(1)/(sqrt3), then (a,b) is equal to
Answer» (-6,11) `(2-(1)/(sqrt3),2+(1)/(sqrt3))` (11,-6) (6,11) Answer :C

Discussion

5728.	If X is a poisson variate such that P(X = 0) = P(X = 1) then find the parameter lambda.
Answer» ANSWER :2

Discussion

5729.	For what value of k, the function defined by {:(f(x),=(log(1+2x)sinx^(6))/(x^(3)),",","for" x != 0),(,=k, ",","for"x = 0):} is continuous at x = 0?
Answer» 2 `1/2` `(PI)/90` `90/pi` ANSWER :C

Discussion

5730.	A differentiablefunctionf Is satifyingthe relationf(x+y)= f(x) + f(y) + 2xy( x+y) -(1)/(3)AA x, y in R andlim_( h to 0)( 3f(h)-1)/(6h)=(2)/(3). Then the valueof[f(2)]is( where [x] represents thegreatest integer function )_________.
Answer» ANSWER :8

Discussion

5731.	If PQ is a focal chord of the ellipse (x ^(2))/(25) + (y ^(2))/( 16 ) =1 which passes through S -= (3,0) and PS =2 then length of chord PQ is
Answer» ANSWER :10

Discussion

5732.	Any point in half plane 2x+3y-12ge0 is
Answer» (0,-12) (12,0) (4,0) (0,-4) ANSWER :B

Discussion

5733.	Find the principal value of cos^-1(-frac(1)(2))
Answer» SOLUTION :`cos(pi-pi/3)=-cosfracpi3=-1/2 `and `pi-pi/3=(2pi)/3IN[0,pi]` `therefore`The principal VALUE of `cos^(-1)(-1/2)=(2pi)/3`

Discussion

5734.	The sum of the fourth powers of the roots of the equation x^(4) - x^(3) - 7x^(2) + x + 6= 0is
Answer» 99 `-140` `-99` `140` ANSWER :1

Discussion

5735.	Each question given below has four possible answers, out of which only one is correct. Choose the correct one. (2hati-4hatj).(hati+hatj+hatk) = ______
Answer» -3 2 -1 -2 Answer :D

Discussion

5736.	If the straight lines ax+by+c=0 and x cos alpha+y sin alpha=c, enclose an angle pi//4 between them and meet the straight line x sin alpha-y cos alpha=0 in the same point, then
Answer» `a^(2)+B^(2)=C^(2)` `a^(2)+b^(2)=2` `a^(2)+b^(2)=2c^(2)` `a^(2)+b^(2)=4` ANSWER :B

Discussion

5737.	Evaluate: int(cos2x)/(sinx+cosx)^2dx
Answer» `(-1)/(sinx+cosx)+C` `logabs(sinx+cosx)+C` `logabs(sinx-cosx)+C` `1/((sinx+cosx)^(2))` ANSWER :B

Discussion

5738.	Total number of solutions of the equationsin pi x=\|ln_(e)\|x\|\| is :
Answer» 8 10 9 6 Answer :D

Discussion

5739.	There are two groups of subjects one of which consists of 5 Science and 3 Engineering subjects and the other consists of 3 Science and 5 Engineering subjects. An unbiased die is cost. If 3 or 5 turns up, a subject is selected at random from the first group, other wise the subject is selected at random from the second group. The probability that an Engineering subject is selected ultimatly is
Answer» `13//16` `13//24` `13//26` `7//16` ANSWER :B

Discussion

5740.	Evaluate the following : [[8,-1,-8],[-2,-2,-2],[3,-5,-3]]
Answer» Solution :`[[8,-1,-8],[-2,-2,-2],[3,-5,-3]]` `-2[[8,-1,-8],[1,1,1],[3,-5,-3]]` `(-2)[[9,7,-8],[0,0,1],[8,-2,-3]]` (Replacing `C_1` and `C_2` by `C_1-C_2` and `C_2-C_3` RESPECTIVELY) `-2 {-1[[9,7],[8,-2]]}` = 2(-18-56)=2(-74)=-148

Discussion

5741.	Let a=hat(i)+hat(j)+hat(k), b=-hat(i)+hat(j)+hat(k), c=hat(i)-hat(j)+hat(k) and d=hat(i)+hat(j)-hat(k). Then, the line of intersection of planes one determined by a, b and other determined by c, d is perpendicular to
Answer» X-axis Y-axis Both X and Y axes Both y and z-axes Answer :D

Discussion

5742.	Find how many number of roots of the equation 1+ sin ^(3)theta = cos ^(6) theta in the inteerval [0,2pi].
Answer» Answer :`=> theta = (3PI)/(2) THEREFORE` FOUR SOLUTIONS in `[0,2pi]`

Discussion

5743.	Evaluate lim_(x to 1) x^((1)/(x-1))
Answer» SOLUTION :N/A

Discussion

5744.	x = sqrt(a^(sin^(-1)t)), y=sqrt(a^(cos^(-1)t)) , Prove that: (dy)/(dx) = -y/x
Answer» ANSWER :`(DY)/(DX) = -y/x`

Discussion

5745.	What is the slope of the tangent to the curve y=3x^2+2x-1 at x=2?
Answer» SOLUTION :`y=3x^2_2x-1rArr(DY)/DX=6x+2` siope of the TANGENT at x=2 is dy)/dx]_((x=2))=14`

Discussion

5746.	int (dx)/(sqrt( x - x^(2)))=
Answer» `2 sin^(-1) sqrt(X) + C ` `2 sin^(-1) x + c ` `2 x sin^(-1)` x + c `sin^(-1) x + c ` ANSWER :A

Discussion

5747.	If the angle theta between the vectors veca=2x^(2)hati+4xhatj+hatk and overset(-)b=7hati-2hatj+xhatk is such that 90^(@) lt theta lt 180^(@), then x lies in the interval
Answer» `(0,1/2)` `(1/2,1)` `(1,3/2)` `(1/2,3/2)` ANSWER :A

Discussion

5748.	A person 2 meters tall casts a shadow 3 meters long. At the same time, a telephone pole casts a shadow 12 meters long, How many meters tall is the pole?
Answer» 4 6 8 11 Answer :C

Discussion

5749.	Form the differential equation by eliminating the arbitrary constant from the equation y = a cos (2x + b)
Answer» `(d^(2)y)/(dx^(2)) + 4Y = 0` `(d^(2)y)/(dx^(2)) - 4y = 0` `(d^(2)y)/(dx^(2)) + 2Y = 0` `(d^(2)y)/(dx^(2)) + y = 0` ANSWER :A

Discussion

5750.	LetA(5,-3), B(2,7) and C(-1,2)be the vertices of a triangleA,B,C. IfPis a point inside the triangle ABCsuch that the triangleAPC,APB and BPChave equal areas, then length of the line segment PBis:
Answer» SOLUTION :Areas of `triangleAPC, triangleAPB,triangleBPC` are EQUAL if P is centroid . `P((5+2-1)/3, (-3+7+2)/3) rArr` P(2,2) Distance PB is 5.

Discussion

Explore topic-wise InterviewSolutions in .

Determine order and degree (if defined) of the following differential equations y'' + (y)^2 + 2y = 0

The points A (2, 3), B(3, 5), C (7, 7) and D(4, 7) are such that

If (x)/((1+x^(2))(3-2x))=(A)/(3-2x)+(Bx+C)/(1+x^(2)) then C=

Write the set of points of continuity of f(x)=|x-1|+|x+1|

A point P (x,y)moves in in xy plane in such a way that sqrt(2) le |x + y| + |x - y| le 3sqrt(2) . Area of region representing all possible position of point P is equal to:

C_(1):x^(2)+y^(2)=r^(2)and C_(2):(x^(2))/(16)+(y^(2))/(9)=1 interset at four distinct points A,B,C, and D. Their common tangents form a peaallelogram A'B'C'D'. If A'B'C'D' is a square, then the ratio of the area of circle C_(1) to the area of circumcircle of DeltaA'B'C' is

If A = (0, 0), B = (3, 0), C = (0, 4) are the vertices of a triangle then match the following {:("I. Centroid",(a)(1,1)),("II. Orthocentre",(b)(1,4//3)),("III. Circumcentre",(c)(0,0)),("IV. Incentre",(d)(4,5)),(,(e)(3//2,2)):}

The value of underset(0)overset(pi//2)int (sin^(3)x)/(sin x + cos x)dxis (pi-1)/(k) . The value of k is ________.

How many of them are divisible by 25

At some integer points a polynomial with integer coefficients take values 1, 2 and 3. Prove that thereexist not more than one integer at which the polynomial is equal to 5.

Let f : R^(+) rarr R satisfies the functional equation f(xy) = e^(xy - x - y) {e^(y) f(x) + e^(x) f(y)}, AA x, y in R^(+). If f'(1) = e, determine f(x).

Choose the correct answer int dx/(x(x^2+1) =

intTan(sqrt((1-x)/(1+x)))dx=

AL, BM and CN are perpendicular from angular points of a triangle ABC on the opposite sides BC, CA and AB respectively. Delta is the area of triangle ABC, (r ) and R are the inradius and circumradius. Radius os the circum circle of Delta LMN is

AL, BM and CN are perpendicular from angular points of a triangle ABC on the opposite sides BC, CA and AB respectively. Delta is the area of triangle ABC, (r ) and R are the inradius and circumradius. If area of Delta LMNis Delta ', then the value of (Delta')/(Delta) is

AL, BM and CN are perpendicular from angular points of a triangle ABC on the opposite sides BC, CA and AB respectively. Delta is the area of triangle ABC, (r ) and R are the inradius and circumradius. If radis of the incircle of Delta LMN is r', then the valur of r' sec A sec B sec C is

AL, BM and CN are perpendicular from angular points of a triangle ABC on the opposite sides BC, CA and AB respectively. Delta is the area of triangle ABC, (r ) and R are the inradius and circumradius. If perimeters of DeltaLMN and Delta ABC an lamda and mu,then the value of (lamda)/(mu) is

For given events A and B if P(A) = 0.25 and P(B) = 0.50 probability that both event occurs together is 0.14. Then ........ is the probability of an event that A and B does not occurs.

Each question given below has four possible answers, out of which only one is correct. Choose the correct one. (hati+hatk)xx(hati+hatj+hatk) = ___

If the function f(x) & g(x) are defined on R to R such that f (x) = {{:( 0 , x in "rational") , (x "," , x in "irrational"):} g(x) = {{:( 0 "," , x in "irrational" ) , ( x "," , x in "rational" ):} then (f-g) (x) is

If f_(1)(x)=1-sqrt(1-x^(2)) and g_(1)'(x)=(sqrt2-1)sqrt(1-x^(2)) for 0le x le 1, then find the area bounded between the curves f_(2)(x)=max{f_(1)(t), 0le tlex}, g_(2)(x)=min{g_(1)(t), 0 le t le x)" for 0le x le 1 and y - axis.

Equation of the plane that contains the lines vecr = (hati +hatj) + lambda(hati + 2hatj - hatk) and vecr = (hati + hatj) + mu(-hati + hatj-2hatk) is

If sum_(r=0)^(n-1)log_(2)((r+2)/(r+1)) prod_(r=10)^(99)log_(r) (r+1), then 'n' is equal to

If f(x)=2x^(3)+mx^(2)-13x+nare 2, 3 roots of theequation f(x)=0, then the value of m and n are

If f is defined in [1,3] by f(x)=x^(3)+bx^(2)+ax such that f(1)-f(3)=0 and f'(c)=0, "where" c=2+(1)/(sqrt3), then (a,b) is equal to

If X is a poisson variate such that P(X = 0) = P(X = 1) then find the parameter lambda.

For what value of k, the function defined by {:(f(x),=(log(1+2x)sinx^(6))/(x^(3)),",","for" x != 0),(,=k, ",","for"x = 0):} is continuous at x = 0?

A differentiablefunctionf Is satifyingthe relationf(x+y)= f(x) + f(y) + 2xy( x+y) -(1)/(3)AA x, y in R andlim_( h to 0)( 3f(h)-1)/(6h)=(2)/(3). Then the valueof[f(2)]is( where [x] represents thegreatest integer function )_________.

If PQ is a focal chord of the ellipse (x ^(2))/(25) + (y ^(2))/( 16 ) =1 which passes through S -= (3,0) and PS =2 then length of chord PQ is

Any point in half plane 2x+3y-12ge0 is

Find the principal value of cos^-1(-frac(1)(2))

The sum of the fourth powers of the roots of the equation x^(4) - x^(3) - 7x^(2) + x + 6= 0is

Each question given below has four possible answers, out of which only one is correct. Choose the correct one. (2hati-4hatj).(hati+hatj+hatk) = ______

If the straight lines ax+by+c=0 and x cos alpha+y sin alpha=c, enclose an angle pi//4 between them and meet the straight line x sin alpha-y cos alpha=0 in the same point, then

Evaluate: int(cos2x)/(sinx+cosx)^2dx

Total number of solutions of the equationsin pi x=|ln_(e)|x|| is :

Evaluate the following : [[8,-1,-8],[-2,-2,-2],[3,-5,-3]]

Let a=hat(i)+hat(j)+hat(k), b=-hat(i)+hat(j)+hat(k), c=hat(i)-hat(j)+hat(k) and d=hat(i)+hat(j)-hat(k). Then, the line of intersection of planes one determined by a, b and other determined by c, d is perpendicular to

Find how many number of roots of the equation 1+ sin ^(3)theta = cos ^(6) theta in the inteerval [0,2pi].

Evaluate lim_(x to 1) x^((1)/(x-1))

x = sqrt(a^(sin^(-1)t)), y=sqrt(a^(cos^(-1)t)) , Prove that: (dy)/(dx) = -y/x

What is the slope of the tangent to the curve y=3x^2+2x-1 at x=2?

int (dx)/(sqrt( x - x^(2)))=

If the angle theta between the vectors veca=2x^(2)hati+4xhatj+hatk and overset(-)b=7hati-2hatj+xhatk is such that 90^(@) lt theta lt 180^(@), then x lies in the interval

A person 2 meters tall casts a shadow 3 meters long. At the same time, a telephone pole casts a shadow 12 meters long, How many meters tall is the pole?

Form the differential equation by eliminating the arbitrary constant from the equation y = a cos (2x + b)

LetA(5,-3), B(2,7) and C(-1,2)be the vertices of a triangleA,B,C. IfPis a point inside the triangle ABCsuch that the triangleAPC,APB and BPChave equal areas, then length of the line segment PBis: