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HOW TO FIND AREA OF THE CIRCLE WHICH IS INTERIOR TO THE PARABOLA

Area Under Curves

Let us write two equations of circle and parabola respectively
4x²+ 4y²= 9 ------------------- (1)

and x²= 4y -------------------(2)

Reducing (1) to standard form by dividing 4 .we get
x²+ y²= (3/2)²
Ist of all draw figures of both the circle and the parabola in cartesian plane.

As it can be seen from figure both curves intersect each other at two points say A and A' .
Next we have to find these two coordinates points of intersection . Solving (1) and (2) to find the values of x and y
Putting the value of ' x²' from (2) in (1) we get

4(4y)+ 4y²= 9

16y + 4y²- 9 = 0

4y²- 16y -9 = 0

y = (-16+20)/8 and (-16-20)/8
y = 1/2 and -9/2

HOW TO FIND AREA OF THE CIRCLE WHICH IS INTERIOR TO THE PARABOLA

So Rejecting the -ve value of y ,because when we put negative value (-9/2) in eq (2) , we shall have two complex values of "x" which are not acceptable.
so only put positive value (1/2) of 'y' in (2) we get two real values of 'x' such that x= 土⇃2,
Now we can write coordinate M(⇃2,0) and N (-⇃2,0)

Required Area = Shaded area
= 2 × Area OBAO
Note this step carefully ↑

Multiplying every terms with 2 which is written at beginning of the previous line.

Putting the values of upper and lower limits of x

ALSO READ HOW TO INTEGRATE INTEGRAL WITH SQUARE ROOT IN NUMERATOR

Final words

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HOW TO FIND THE ANGLE BETWEEN TWO LINES

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HOW TO FIND THE ANGLE BETWEEN TWO LINES WHEN THE EQUATIONS OF GIVEN LINES ARE IN CARTESIAN FORMS

In this post we shall study How to find the angle between two lines ,angle between two lines vectors in Cartesian form, angle between two lines in 3d, angle between two lines calculator, angle between two lines coordinate geometry,derivation of angle between two lines

Problem 1

Consider two lines whose equations are given in cartesian form as

Note that direction Ratios of 1st line is (1 , 2 , 3) and the direction Ratios of 2nd line is (2 , 3 , 4).

Note that Direction Ratios of any line are those numbers written is the denominator in standard form of the equation of the line

we know that if a_1,b_1,c_1 anda_2,b_2,c_2,_{are Direction Ratios of line}L₁_{and Line}L_{2 .}
_{If θ be the angle between two lines ,Then this angle can be formulated as follows}

So putting the values of direction ratios of both the lines in (3) ;

we get

cos θ = [2 + 6 + 12]/sqrt[ 1+ 4+ 9]×sqrt[ 4 +9 +16 ]

cos θ = [20]/√[ 14×29 ];

cos θ = 10/√203

∴ θ = cos^-1 (10/√203)

Hence cos^-1 (10/√203) is the angle between two lines

How to find angle between two line in vector and cartesian forms

Problem 2

Let us consider these two lines in cartesian form

Since these equations of lines are not in standard form , in order to reduce these equations to standard form we shall have to make the coeffs of x, y , z unity.

In 1st part of equation (1) divide the num and den by 2 , and in 2nd part divide num and den by 3.

Similarly divide 2nd part of equation (2) by 5. we can rewrite these equations as follows

After cancellation and simplification , we get equations of lines in standard form

Here Direction Ratios of 1st Line are (2,1 ,3) and Direction Ratios of 2nd Line are (1 ,2 , 4) so using the formula

Putting the values of direction ratios of both the lines in above formula ,we get

cos θ = [2 + 2 + 12]/sqrt[ 4+ 1+ 9]×sqrt[ 1 +4 +16 ]

cos θ = [16]/sqrt[14×21];

cos θ = 16/√294

∴ θ = cos^-1 (16/√294)

Hence cos^-1 (16/√294) is the angle between two lines

Problem 3

How to prove that the given lines are parallel to each others,

To test whether given lines are parallel to each other , just check their direction ratios , id they are proportional to each others , then The given lines would be parallel to each other.

If the Direction ratios of 1st line is (1,2,5) and Direction ratios of 2nd line is (2,4,10) . If we take 2 common from direction ratios of 2nd lines then we have same direction Ratios as that of 1st line hence These lines are parallel o each other.

Problem 4

How to prove that the given lines are Perpendicular to each others. If we calculate the angle between any two lines and it comes out to be 0 ( ZERO ) then these would be perpendicular to each others.

E.g If the D. R. of 1st line are ( 3, -1 , 3 )

And D. R. of 2nd Line are ( -2 ,3 ,3 ).

The cosine of angle between these two line is zero

cos θ = 0

Then θ = 90°

⇒ Lines are Perpendicular to each others

Also Read previous posts
How to Find perpendicular distance between skew lines

How to find slope of line ax+by=c

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To find the angle between two lines

x = y = z and

x = y = -z

Want to check the solution of this problem ?

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Thanks for your precious time to read this post regarding how to find the angle between two lines in Cartesian form.

HOW TO FIND THE PERPENDICULAR DISTANCE BETWEEN TWO SKEW LINES AND PARALLEL LINES

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The Shortest Distance Between Skew Lines

The shortest distance between the lines is the distance which is perpendicular to both the lines given as compared to any other lines that joins these two skew lines.

PERPENDICULAR DISTANCE BETWEEN TWO SKEW LINES

Vector Form

We shall consider two skew lines L1 and L2 and we are to calculate the distance between them. The equations of the given lines are:

Here vector

and vector

are the vectors through which line (1) and (2) passes and

and

are the vectors which are parallel to lines L1 and L2 respectively.

Then perform the following steps

(1) Calculate

(2 ) Calculate

(3) Calculate |

(4 ) Calculate (

−

) . (

)

and if the dot product of these two vectors come out to be negative then take its absolute value as distance can not be a negative quantity .

(5 ) Put all these values in the formula given below and the value so calculated is the shortest distance between two skew Lines.

⃗⃗

Problem : How to find the shortest distance between two skew lines in vector form whose equations are given by

Now write the values of

and

Now find out the difference of

and

HOW TO FIND THE PERPENDICULAR DISTANCE BETWEEN TWO SKEW LINES AND PARALLEL LINES

After solving this determinant and little simplification we get ,

The magnitude of this vector

= √(169+64+4)=√(237)

Now find Dot Product (

) and (

) ,

= (0)(-13)+(-6)(8)+(5)(-2)

= -48 -10

= -58

Taking the absolute value of

| (

−

) . (

) | = 58

Now putting all these values in SD Formula written above , we can have

SD = 58/(√237) Units

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The Shortest Distance Between Parallel Lines

Consider two parallel lines

Here vector

and vector

are the vectors through which line (1) and (2) passes and

is the vector which is parallel to both lines L1 and L2 respectively. Now perform the following steps

( 1 ) Calculate

( 2 ) Calculate |

( 3 ) Calculate (

) ×

Put all these values in the formula given below and the value so calculated is the shortest distance between two Parallel Lines, and if it comes to be negative then take its absolute value as distance can not be negative

Problem 2 : How to find the shortest distance between two Parallel lines in vector form whose equations are given by

As we know these are parallel line because both these equations are parallel to same vector -2i +3J+5k.

Now write the values of

, and

as follows

Now find out the difference of

and

Now Calculate (

) ×

After solving this determinant and simplification we get ,

The magnitude of this vector is

= √2069

Similarly the magnitude of vector

is √38

SD = √(2069 /38) Units

At Last

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