Visual SolutionTricky

Question

The angle between the tangents drawn from the origin to the circle

x^2 + y^2 - 4x - 4y + 4 = 0

is:

(A)

\dfrac{\pi}{2}

(B)

\dfrac{\pi}{3}

(C)

\dfrac{\pi}{4}

(D)

\dfrac{\pi}{6}

Solution Path

Center (2,2), r=2, distance from origin = 2*sqrt(2). Tangent length = 2. Half-angle tan = r/L = 1, so angle = pi/2. Origin lies on the director circle.

01Question Setup

1/4

Find the angle between the tangents from the origin to the circle

x^2+y^2-4x-4y+4=0

\alpha = \;?

02Circle Properties

2/4

Center

= (2,2)

, radius

= 2

. Distance from origin to center

= 2\sqrt{2}

. Tangent length

= \sqrt{S_1} = 2

d = 2\sqrt{2},\; r = 2,\; L = 2

03Compute the AngleKEY INSIGHT

3/4

Half-angle formula:

\tan(\alpha/2) = r/L = 2/2 = 1

. So

\alpha/2 = \pi/4

, giving

\alpha = \pi/2

. The origin lies on the director circle since

d^2 = 2r^2

\alpha = \dfrac{\pi}{2}

04Final Answer

4/4

Angle between tangents

= \pi/2

. When a point lies on the director circle (

d^2 = 2r^2

), the tangent pair is always perpendicular.

\boxed{\dfrac{\pi}{2}}

Concepts from this question2 concepts unlocked

★

Tangent from an External Point

STANDARD

From a point outside a circle, exactly two tangent lines can be drawn. For the circle x^2 + y^2 = a^2, a tangent with slope m has the equation y = mx +/- a*sqrt(1 + m^2). The length of each tangent equals sqrt(S1). The chord of contact joining the two tangent points has equation T = 0.

y = mx \pm a\sqrt{1 + m^2}, \quad L = \sqrt{S_1}

Tangent problems are the most frequently tested subtopic in Circles. JEE tests tangent equations, tangent lengths, and the number of common tangents between two circles. Missing the +/- sign is the most common error.

Tangent with given slopeTangent from external pointCommon tangents to two circlesDirector circle

Practice (12 Qs) →

Power of a Point

EASY

The power of a point P(x1, y1) with respect to a circle S: x^2 + y^2 + 2gx + 2fy + c = 0 is defined as S1 = x1^2 + y1^2 + 2gx1 + 2fy1 + c. It equals (distance from center)^2 - r^2. If P is outside, S1 > 0 and equals the square of the tangent length. If P is on the circle, S1 = 0. If P is inside, S1 < 0.

S_1 = x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c = d^2 - r^2

Power of a point unifies tangent length, position of a point, and radical axis into one concept. JEE frequently tests whether students can correctly determine if a point is inside, outside, or on a circle using S1.

Length of tangentPosition of point relative to circleRadical axisChord of contact

Practice (10 Qs) →

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