Conic Sections MATHS - Conic Sections

The equation of the circle passing through (3, 6) and whose centre is (2, -1) is:

The equation of the incircle formed by the coordinate axes and the line 4x + 3y = 6 is:

The equation of the parabola whose vertex is (a, 0) and the directrix has the equation x + y = 3a, is

Find the equation of a circle with center (0, 0) and radius 5:

If the point (2, k) lies outside the circles x² + y² + x - 2y - 14 = 0 and x² + y² = 13 then klies in the interval:

State Whether the statements are True or False. Justify.
The line x + 3y = 0 is a diameter of the circle x² + y² + 6x + 2y = 0.

State Whether the statements are True or False. Justify.
The shortest distance from the point (2, -7) to the circle x² + y² - 14x - 10y - 151 = 0 is equal to 5.
[Hint: The shortest distance is equal to the difference of the radius and the distance between the centre and the given point]

State Whether the statements are True or False. Justify.
The line lx + my + n = 0 will touch the parabola y² = 4ax if ln = am²

State Whether the statements are True or False. Justify.
If the line lx + my = 1 is a tangent to the circle x² + y² = a², then the point (l, m) lies on a circle.
[Hint: Use that distance from the centre of the circle to the given line is equal to radius of the circle]

Fill in the blank.
The equation of the ellipse having foci (0, 1), (0, -1) and minor axis of length 1 is _____________.

Fill in the blank.
The equation of the parabola having focus at (-1, -2) and the directrix x - 2y + 3 = 0 is _____________.

Fill in the blank.
The equation of the circle having centre at (3, -4) and touching the line 5x + 12y - 12 = 0 is __________.
[Hint: To determine radius find the perpendicular distance from the centre of the circle to the line]

Fill in the blank.
The equation of the circle circumscribing the triangle whose sides are the lines y = x + 2, 3y = 4x, 2y = 3x is ___________.

Find the equation of the circle with centre (-3, 2) and radius 4

Find an equation of the circle with centre at (0,0) and radius r.

An equilateral triangle is inscribed in the parabola y² = 4ax where one vertex is at the vertex of the parabola. Find the length of the side of the triangle.

Find the area of the triangle formed by the lines joining the vertex of the parabola x² = 12y to the ends of its latus rectum.

If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus.

A man running a racecourse notes that the sum of the distances from the two flag posts from him is always 10 m and the distance between the flag posts is 8 m. Find the equation of the path traced by the man.

A rod of length 12 m moves with its ends always touching the coordinates axes. Determine the equation of the locus of a point P on the rod, which is 3 cm from the end in contact with the X-axis.

An arc is in the form of a semi-ellipse. It is 8 m wide and 2 m high at the centre. Find the height of the arch at a point 1.5 m from one end.

The cable of a uniformly loaded suspension bridge hangs in the form of a parabola. The roadway which is horizontal and 100 m long is supported by vertical wires attached to the cable, the longest wire being 30 m and the shortest being 6 m. Find the length of a supporting wire attached to the roadway 18 m from the middle.

An arc is in the form of a parabola with its axis vertical. The arch is 10 m high and 5 m wide at the base. How wide is it 2 m from the vertex of the parabola?

Directions: In the following questions, the Assertions (A) and Reason(s) (R) have been put forward. Read both the statements carefully and choose the correct alternative from the following:
If the distances of foci and vertex of hyperbola from the centre are c and a respectively, then
Assertion: A line through the focus and perpendicular to the directrix is called the x - axis of the parabola.
Reason: The point of intersection of parabola with the axis is called the vertex of the parabola.

1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
2. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
3. Assertion is correct statement but Reason is wrong statement.
4. Assertion is wrong statement but Reason is correct statement.

Directions: In the following questions, the Assertions (A) and Reason(s) (R) have been put forward. Read both the statements carefully and choose the correct alternative from the following:
If the distances of foci and vertex of hyperbola from the centre are c and a respectively, then
Assertion: Eccentricity is always less than 1.
Reason: Foci are at a distance of ae from the centre.

1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
2. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
3. Assertion is correct statement but Reason is wrong statement.
4. Assertion is wrong statement but Reason is correct statement.

Directions: In the following questions, the Assertions (A) and Reason(s) (R) have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Parabola is symmetric with respect to the axis of the parabola.
Assertion: If the equation of standard parabola has a term y², then the axis of symmetry is along the x - axis.
Reason: If the equation of standard parabola has a term x², then the axis of symmetry is along the x - axis.

1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
2. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
3. Assertion is correct statement but Reason is wrong statement.
4. Assertion is wrong statement but Reason is correct statement.

Directions: In the following questions, the Assertions (A) and Reason(s) (R) have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion: The sum of focal distances of a point on the ellipse 9x² + 4y² - 18x - 24y + 9 = 0 is 4.
Reason: The equation 9x² + 4y² - 18x - 24y + 9 = 0 can be expressed as 9(x -1)² + 4(y - 3)² = 36.

1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
2. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
3. Assertion is correct statement but Reason is wrong statement.
4. Assertion is wrong statement but Reason is correct statement.

Directions: In the following questions, the Assertions (A) and Reason(s) (R) have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion: The length of major and minor axes of the ellipse 5x² + 9y² - 54y + 36 = 0 are 6 and 10, respectively.
Reason: The equation 5x² + 9y² - 54y + 36 = 0 can be expressed as 5x² + 9(y - 3)² = 45.

1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
2. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
3. Assertion is correct statement but Reason is wrong statement.
4. Assertion is wrong statement but Reason is correct statement.