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R

Rakhi 08 Nov in  Class XI-XII Tuition (PUC), Mathematics(Class XI-XII Tuition (PUC)), Conic Sections

Show that the point (5, 6) outside the circle: (square of x) + (square of y) = 4?

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Sarvajeet Kumar replied | 3 days ago

S'=(square of 5) + (square of 6) - 4 = -57.

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Tapas replied | 2 days ago

The equation of the circle is:
x^2 + y^2 = 4 = 2^2.
Therefore the radius of the circle is 2 units with its center at (0, 0). So, any point at a distance bigger than 2 units must be outside the circle.
Distance between the center (0, 0) and point (5, 6) is:
Sq. root of [(5-0)^2 + (6-0)^2] = sq root of (25 + 36) = sq root of 61 = 7.8. This is greater than the radius...  more»
The equation of the circle is:
x^2 + y^2 = 4 = 2^2.
Therefore the radius of the circle is 2 units with its center at (0, 0). So, any point at a distance bigger than 2 units must be outside the circle.
Distance between the center (0, 0) and point (5, 6) is:
Sq. root of [(5-0)^2 + (6-0)^2] = sq root of (25 + 36) = sq root of 61 = 7.8. This is greater than the radius of the circle. So the point (5, 6) is outside the circle. «less

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R

Raj 08 Nov in  Class XI-XII Tuition (PUC), Mathematics(Class XI-XII Tuition (PUC)), Conic Sections

Find the equation of the circle which passes through the points (1, 2), (2, 2) and (4, -1)?

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Sarvajeet Kumar replied | 3 days ago

square of (x-3/2) + square of (y-3/2) = 1/2.

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Tapas replied | 2 days ago

Take the general equation for a circle:
x^2 + y^2 + 2gx + 2fy + c = 0, where the center is at (-g, -f) and the radius is sq root of (g^2 +f^2 -c).
Now use the given points to get three equations to solve for three unknowns, g, f, and c. The equations with point (1, 2) becomes:
1^2 + 2^2 + 2xgx1 + 2xfx2 + c = 0
=> 2g + 4f + c = - 5 .............. (1)
Similarly,...  more»
Take the general equation for a circle:
x^2 + y^2 + 2gx + 2fy + c = 0, where the center is at (-g, -f) and the radius is sq root of (g^2 +f^2 -c).
Now use the given points to get three equations to solve for three unknowns, g, f, and c. The equations with point (1, 2) becomes:
1^2 + 2^2 + 2xgx1 + 2xfx2 + c = 0
=> 2g + 4f + c = - 5 .............. (1)
Similarly, the equation with the point (2, 2) is:
=> 4g + 4f + c = - 8 .............. (2)
And, the equation with the point (4, -1) is:
=> 8g - 2f + c = - 17 .............. (3)

From (1) - (2) we get: -2g = 3 --> g = -3/2.
Using this value of g in (1) we get:
4f + c = - 2 ............ (4)
Using this value of g in (3) we get:
-2f + c = - 5 ............ (5)
From (4) - (5) we can get:
6f = 3 --> f = 1/2.
Using this value of f in (5) we get: c = - 4.
So, the equation of the circle is:
x^2 + y^2 + 2(-3/2)x + 2(1/2)y + (- 4) = 0.
After simplification:
--> x^2 + y^2 -3x +y - 4 = 0 (Answer) «less

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S

Sriram 08 Nov in  Class XI-XII Tuition (PUC), Mathematics(Class XI-XII Tuition (PUC)), Conic Sections

Find the distance between the directrices of the ellipse: square of (x) / 36 + square of (y) / 20 = 1?

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Sarvajeet Kumar replied | 3 days ago

18.

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Answer

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S

Selvi 08 Nov in  Class XI-XII Tuition (PUC), Mathematics(Class XI-XII Tuition (PUC)), Conic Sections

Find the equation of the hyperbola with eccentricity 3 / 2 and foci at (2, 0)?

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Sarvajeet Kumar replied | 3 days ago

square of (x) / (16/9) - square of (y) / (20/9) = 1.

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N

Nasreen 08 Nov in  Class XI-XII Tuition (PUC), Mathematics(Class XI-XII Tuition (PUC)), Conic Sections

Define a circle as the locus of a point?

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Hussam Uddin replied | 22 Nov

Circle is locus of a point , whose distance from a fixed point i.e centre is constant i.e radius

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Sarvajeet Kumar replied | 3 days ago

Locus of a point whose distance from a fixed point is remain constant.

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B

Bivash 08 Nov in  Class XI-XII Tuition (PUC), Mathematics(Class XI-XII Tuition (PUC)), Conic Sections

Find the centre and the radius of the circle from the given equation: (square of x) + (square of y) = 16?

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Tapas replied | 28 Nov

Circle: x^2 + y^2 = 16 = 4^2.
Center of the circle: (0, 0). Radius = 4 units.

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Sarvajeet Kumar replied | 3 days ago

The center = (0, 0), Radius = 4.

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N

Neetu 08 Nov in  Class XI-XII Tuition (PUC), Mathematics(Class XI-XII Tuition (PUC)), Conic Sections

Find the equation of the circle whose centre is (3, 6) and which passes through the point (-1, 1)?

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Sarvajeet Kumar replied | 3 days ago

Square of (x-3) + Square of (x-6) = 41.

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N

Neha 08 Nov in  Class XI-XII Tuition (PUC), Mathematics(Class XI-XII Tuition (PUC)), Conic Sections

Find the whether (1, 3) lies inside, or outside or on the circle (square of x) + (square of y) = 16?

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Gunturi replied | 3 days ago

Substitute those values in the L.H.S of the equation. if
LHS < RHS --- Lies Inside
LHS = RHS ---- Lies on the circle
LHS > RHS ---- Lies outside of the circle.

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Sarvajeet Kumar replied | 3 days ago

Inside.

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H

Harsh 08 Nov in  Class XI-XII Tuition (PUC), Mathematics(Class XI-XII Tuition (PUC)), Conic Sections

Define latus rectum of a Parabola?

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Sarvajeet Kumar replied | 3 days ago

The latus rectum of a parabola, square of y =4ax is 4a.

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H

Himanshu 08 Nov in  Class XI-XII Tuition (PUC), Mathematics(Class XI-XII Tuition (PUC)), Conic Sections

Find the axis of the parabola: (square of y) = x?

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Sarvajeet Kumar replied | 3 days ago

The x axis i.e., y = 0.

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S

Sammeyka 08 Nov in  Class XI-XII Tuition (PUC), Mathematics(Class XI-XII Tuition (PUC)), Conic Sections

Find the equation of the line which passes through the focus of the Parabola: (square of y) = 20x?

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Sarvajeet Kumar replied | 3 days ago

There are infinite number of lines at different slopes (including its axis) which pass through the focus (5,0) of the parabola, (square of y) = 20x. So, you tell me either one slope or one point.

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S

Swati 08 Nov in  Class XI-XII Tuition (PUC), Mathematics(Class XI-XII Tuition (PUC)), Conic Sections

Find the equation of the circle having (1, -2) as its centre and passing through 3x + y = 14, 2x + 5y = 18?

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Hussam Uddin replied | 22 Nov

Equation of circle is (x-1)2 + (y +2)2 = r2 , we need to find r , point of intersection of 2 lines is 13x = 52, x= 4 , y = 2, therefore r2 = 25.

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A

Alby 08 Nov in  Class XI-XII Tuition (PUC), Mathematics(Class XI-XII Tuition (PUC)), Conic Sections

Show that the point (x, y) given by: x = 2at / (1 + square of t) and y = a (1 - square of t) / 1 + t lies on a circle for all real values of t such that - 1 < t < 1 where a is any given real numbers?
If a circle passes through the point (0, 0) (a, 0) (0, b) then find the coordinates of its circle?

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T

Thinagaran 08 Nov in  Class XI-XII Tuition (PUC), Mathematics(Class XI-XII Tuition (PUC)), Conic Sections

Find the equation of the circle which touches x - axis and whose centre is (1, 2)?

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G

Gagandeep 08 Nov in  Class XI-XII Tuition (PUC), Mathematics(Class XI-XII Tuition (PUC)), Conic Sections

If the lines 3x - 4y + 4 = 0 and 6x - 8y - 7 = 0 are tangents to a circle, then find the radius of the circle?

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M

Mukesh 08 Nov in  Class XI-XII Tuition (PUC), Mathematics(Class XI-XII Tuition (PUC)), Conic Sections

If one end of a diameter of the circle (square of x) + (square of y) - 4 - 6y + 11 = 0 is (3, 4), then find the coordinate of the other end of the diameter?
Find the equation of the circle having (1, -2) as its centre and passing through 3x + y = 14, 2x + 5y =18?

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M

Madhavan 08 Nov in  Class XI-XII Tuition (PUC), Mathematics(Class XI-XII Tuition (PUC)), Conic Sections

Find the equation of the circle which passes through the points (1, 3), (2, 3) and whose centre lies on the line 2x + 3y - 1 = 0?

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A

Anand 08 Nov in  Class XI-XII Tuition (PUC), Mathematics(Class XI-XII Tuition (PUC)), Conic Sections

Prove that the length of the latus rectum of the Parabola: (square of y) = 4ax is 4a?

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B

Bala 08 Nov in  Class XI-XII Tuition (PUC), Mathematics(Class XI-XII Tuition (PUC)), Conic Sections

Define the equation of the parabola in the standard form: (square of y) = 4ax?

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A

Aishwarya 08 Nov in  Class XI-XII Tuition (PUC), Mathematics(Class XI-XII Tuition (PUC)), Conic Sections

Find the equation of the Parabola which is symmetric about y - axis, vertex origin and which passes through (6, -8)?

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