All BASICSTANDARDADVANCED

Question 251

For each natural number k, let Ck denote the circle with radius k centimeters and center at the origin. On the circle Ck, a particle moves k centimeters in the counterclockwise direction. After completing its motion on Ck the particle moves to Ck + 1 in the radial direction. The motion of the particle continues in this manner. The particle starts at ( 1, 0 ). If the particle crosses the positive direction of the X–axis for the first time on the circle Cn then n = . . . . .

IIT 1997

My Self Assessment

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Let  be a circle. A pair of tangents from (4, 5) and a pair of radii form a quadrilateral of area . . . . .

The equation of the line passing through the points of intersection of the circles
 and
 is . . . . .

The area of the triangle formed by the tangents from the point (4, 3) to the circle  and the line joining their point of contact is .

A polygon of nine sides, each of length 2, is inscribed in a circle. The radius of the circle is . . . . .

If the triangle  another circle C2 of radius 5 in such a manner that the common chord is of maximum length and a slope equal to  , then the coordinates of the centre of C2 are . . . . .

ABC is an isosceles triangle in a circle of radius r. If AB = AC and h is the altitude from A to BC then the triangle ABC has perimeter , area A = . . . . .

Also  . . . . .

The area of triangle formed by the positive X–axis and the normal and tangent to the circle  at  is . . . . . .

A circle passes through the point of intersection of the coordinate axes with the lines  and x , then λ = . . . . .

The equation of the locus of the midpoints of the chord of the circle  that subtends an angle of  at the centre is  . . . . .

Intercepts on the line y = x by the circle  is AB. Equation of the circle with AB as diameter is  . . . . .

For each natural number k, let Ck denote the circle with radius k centimeters and center at the origin. On the circle Ck, a particle moves k centimeters in the counterclockwise direction. After completing its motion on Ck the particle moves to Ck + 1 in the radial direction. The motion of the particle continues in this manner. The particle starts at ( 1, 0 ). If the particle crosses the positive direction of the X–axis for the first time on the circle Cn then n = . . . . .

The chord of contact of the pair of tangents drawn from each point on the line  to the circle  passes through the point . . . . .

Two vertices of an equilateral triangle are (- 1, 0) and (1, 0) and its third vertex lies above the X–axis, the equation of circumcircle is . . .

If  is a normal to  then k is

a) 3

b) 9

c) – 9

d) – 3

The equation of the directrix of the parabola y2 + 4y + 4x +2 = 0 is

a) x = − 1

b) x = 1

c)

d)

The equation of the common tangent touching the circle
 and the parabola , above X–axis is

a)

b)

c)

d)

The equation of the common tangent to the curves  and  is

a)

b)

c)

d)

The focal chord of  is tangent to  then the possible value of the slope of this chord are

a)   

b)  

c)   

d)   

The angle between the tangents drawn from the point (1, 4) to the parabola  is

a)

b)

c)

d)

The axis of the parabola is along the line  and the distance of the vertex and focus from origin are  and  respectively. If vertex and focus both lie in the first quadrant, then the equation of the parabola is

a)

b)

c)

d)

(Multiple choice)

The equation of common tangent to the parabolas  and  is/are

a)

b)

c)

d)

Suppose that the normals drawn at three different points on the

parabola  pass through the point (h, 0). Show that h > 2.

A is a point on the parabola . The normal at A cuts the parabola again at B. If AB subtends a right angle at the vertex of the parabola, find the slope of AB.

Find the equation of the normal to the curve  which passes through the point (1, 2).