Test: Comprehension Based Questions: Circle
8 Questions MCQ Test Maths 35 Years JEE Main & Advanced Past year Papers | Test: Comprehension Based Questions: Circle
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PASSAGE-1
ABCD is a square of side length 2 units. C1 is the circle touching all the sides of the square ABCD and C2 is the circumcircle of square ABCD. L is a fixed line in the same plane and R is a fixed point.
Q. If P is any point of C1 and Q is another point on C2, then
is equal to (2006 - 5M, –2)
ABCD is a square of side length 2 units. C1 is the circle touching all the sides of the square ABCD and C2 is the circumcircle of square ABCD. L is a fixed line in the same plane and R is a fixed point.
is equal to (2006 - 5M, –2) Without loss of generality we can assume the square ABCD with its vertices A (1, 1), B (–1, 1), C (–1, –1), D (1, –1)
P to be the point (0, 1) and Q as (
,0 ).
Then, 
PASSAGE-1
ABCD is a square of side length 2 units. C1 is the circle touching all the sides of the square ABCD and C2 is the circumcircle of square ABCD. L is a fixed line in the same plane and R is a fixed point.
Q. If a circle is such that it touches the line L and the circle C1 externally, such that both the circles are on the same side of the line, then the locus of centre of the circle is (2006 - 5M, –2)
ABCD is a square of side length 2 units. C1 is the circle touching all the sides of the square ABCD and C2 is the circumcircle of square ABCD. L is a fixed line in the same plane and R is a fixed point.
Let C' be the said circle touching C1 and L, so that C1 and C' are on the same side of L. Let us draw a line T parallel to L at a distance equal to the radius of circle C1, on opposite side of L.
Then the centre of C' is equidistant from the centre of C1 and from line T.
⇒ locus of centre of C' is a parabola.
PASSAGE-1
ABCD is a square of side length 2 units. C1 is the circle touching all the sides of the square ABCD and C2 is the circumcircle of square ABCD. L is a fixed line in the same plane and R is a fixed point.
Q. A line L' through A is drawn parallel to BD. Point S moves such that its distances from the line BD and the vertex A are equal. If locus of S cuts L' at T2 and T3 and AC at T1, then area of ΔT1T2T3 is (2006 - 5M, –2)
ABCD is a square of side length 2 units. C1 is the circle touching all the sides of the square ABCD and C2 is the circumcircle of square ABCD. L is a fixed line in the same plane and R is a fixed point.
Since S is equidistant form A and line BD, it traces a parabola. Clearly, AC is the axis, A (1, 1) is the focus
and 
is the vertex of parabola.
T2 T3 = latus rectum of parabola
∴ Area (ΔT1T2T3) =
PASSAGE-2
A circle C of radius 1 is inscribed in an equilateral triangle PQR.
The points of contact of C with the sides PQ, QR, RP are D, E, F, respectively. The line PQ is given by the equation
and the point D is
Further, it is given that the origin and the centre of C are on the same side of the line PQ.
Q. The equation of circle C is (2008)
Slope of CD = 
∴ Parametric equation of CD is
∴ Two possible coordinates of C are
or 
As (0, 0) and C lie on the same side of PQ
should be the coordinates of C.
NOTE THIS STEP : Remember (x1, y1) and (x2, y2) lie on the same or opposite side of a line ax + by + c = 0 according as 
∴ Equation of the circle is
PASSAGE-2
A circle C of radius 1 is inscribed in an equilateral triangle PQR.
The points of contact of C with the sides PQ, QR, RP are D, E, F, respectively. The line PQ is given by the equationand the point D is
Further, it is given that the origin and the centre of C are on the same side of the line PQ.
Q. Points E and F are given by (2008)
ΔPQR is an equilateral triangle, the incentre C must coincide with centriod of ΔPQR and D, E, F must concide with the mid points of sides PQ, QR and RP respectively.
Also 
Writing the equation of side PQ in symmetric form we
get, 
∴ Coordinates of P = 
and
coordinates of 
Let coordinates of R be (α,β) , then using the formula for centriod of Δ we get
⇒ α = 0 and β = 0
∴ Coordinates of R = (0, 0)
Now coordinates of E = mid point of QR =
and coordinates of F = mid point of PR = 
PASSAGE-2
A circle C of radius 1 is inscribed in an equilateral triangle PQR.
The points of contact of C with the sides PQ, QR, RP are D, E, F, respectively. The line PQ is given by the equationand the point D is
Further, it is given that the origin and the centre of C are on the same side of the line PQ.
Q. Equations of the sides QR, RP are (2008)
Equation of side QR is y = 
and equation of side RP is y = 0
Paragraph 3 Given the implicit function y3 – 3y + x = 0
For x ∈(–∞, –2) ∪ (2,∞) it is y = f (x) real valued differentiable function and for x ∈ (–2, 2) it is y = g(x) real valued differentiable function.
PASSAGE-3
A tangent PT is drawn to the circle x2 + y2 = 4 at the point 
. A straight line L, perpendicular to PT is a tangent to the circle (x – 3)2 + y2 = 1. (2012)
Q. A possible equation of L is
Equation of tangent PT to the circle x2 + y2 = 4 at the point 
Let the line L, perpendicular to tangent PT be
As it is tangent to the circle (x – 3)2 + y2 = 1
∴ length of perpendicular from centre of circle to the tangent = radius of circle.
= – 1 or – 5
∴ Equation of L can be 
PASSAGE-3
A tangent PT is drawn to the circle x2 + y2 = 4 at the point . A straight line L, perpendicular to PT is a tangent to the circle (x – 3)2 + y2 = 1. (2012)
Q. A common tangent of the two circles is
From the figure it is clear that the intersection point of two direct common tangents lies on x-axis.
Also ΔPT1C1 ~ ΔPT2C2 ⇒ PC1 : PC2 = 2 : 1
or P divides C1C2 in the ratio 2 : 1 externally
∴ Coordinates of P are (6, 0) Let the equation of tangent through P be y = m (x – 6)
As it touches x2 + y2 = 4
36 m2 = 4(m2 + 1)
∴ Equations of common tangents are
Also x = 2 is the common tangent to the two circles.
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