Section A — MCQ (Single Correct)
Question 1
Find the equation of the circle passing through the origin $(0,0)$ that has intercepts equal to $a = 4$ and $b = 6$ on the positive x and y axes respectively.
A
$x^2 + y^2 - 4x - 6y = 0$
B
$x^2 + y^2 + 4x + 6y = 0$
C
$x^2 + y^2 - 2x - 3y = 0$
D
$x^2 + y^2 - 8x - 12y = 0$
Question 2
Find the equation of the tangent line to the circle $x^2 + y^2 = 25$ at the point $P(-3, 4)$ lying on its boundary.
A
$-3x + 4y = 25$
B
$3x + 4y = 25$
C
$-3x + 4y = 5$
D
$4x + 3y = 25$
Question 3
Calculate the length of the tangent line segment drawn from the external point $P(5, 6)$ to the circle curve $x^2 + y^2 = 9$.
A
$\sqrt{52}$
B
$4$
C
$2\sqrt{13}$
D
$\sqrt{32}$
Question 4
Find the equation of the chord of contact drawn from the external point $P(2, 3)$ to the circle curve $x^2 + y^2 = 4$.
A
$2x + 3y = 4$
B
$2x + 3y = 0$
C
$3x + 2y = 4$
D
$2x - 3y = 4$
Question 5
Find the equation of the common chord for the two intersecting circles $x^2 + y^2 - 4x - 2y + 1 = 0$ and $x^2 + y^2 - 2x - 4y + 1 = 0$.
A
$x - y = 0$
B
$x + y = 0$
C
$x - y + 1 = 0$
D
$2x + 2y = 1$
Question 6
Find the value of the parameter $k$ for which the two circles $x^2 + y^2 - 2x - 4y + 1 = 0$ and $x^2 + y^2 - 4x + ky + 3 = 0$ intersect orthogonally.
A
$-3$
B
$-1$
C
$2$
D
$1$
Question 7
Determine the total number of common tangents that can be drawn to the two circles $x^2 + y^2 = 4$ and $x^2 + y^2 - 8x - 6y + 21 = 0$.
A
$4$
B
$3$
C
$2$
D
$1$
Question 8
Find the equation of the normal line to the circle $x^2 + y^2 = 16$ at the boundary point $P(2, 2\sqrt{3})$.
A
$y = \sqrt{3}x$
B
$y = -\sqrt{3}x$
C
$x = \sqrt{3}y$
D
$\sqrt{3}x + y = 4$
Question 9
Find the equation of the chord of the circle $x^2 + y^2 = 16$ whose midpoint is located at $M(1, 2)$.
A
$x + 2y = 5$
B
$x + 2y = 16$
C
$2x + y = 5$
D
$x - 2y = 5$
Question 10
Find the radius of the circle given parametrically by $x = 2 + 5\cos\theta$ and $y = -3 + 5\sin\theta$.
A
$5$
B
$25$
C
$\sqrt{5}$
D
$2$
Section B — Integer Type
Question 11 — Integer answer
If the line $y = x + c$ is tangent to the circle $x^2 + y^2 = 18$, find the positive value of the constant intercept parameter $c$.
Question 12 — Integer answer
Find the total number of common tangents that can be drawn to two circles that intersect each other at exactly two distinct points.
Question 13 — Integer answer
If the orthogonal system condition satisfies $2g_1 g_2 + 2f_1 f_2 = c_1 + k$, and the second circle has a constant term $c_2 = 5$ with $c_1 = 4$, find the integer value of $k$.
Section C — Assertion & Reasoning
Question 14 — Assertion / Reason
Assertion (A): The straight line equation $3x - 4y + 25 = 0$ stands as a valid tangent line to the circle $x^2 + y^2 = 25$.Reason (R): A straight line is tangent to a circle if and only if its perpendicular distance from the center point equals the metric radius.
A
Both A and R are true and R is the correct explanation of A
B
Both A and R are true but R is NOT the correct explanation of A
C
A is true but R is false
D
A is false but R is true
Solution: Both A and R are true, and R is the correct explanation.
Question 15 — Assertion / Reason
Assertion (A): The chord of contact from an internal point $P(1,1)$ to the circle $x^2 + y^2 = 25$ can be found using the formula $T = 0$.Reason (R): A chord of contact can only be generated from an external point lying outside the circle boundary, where $S_1 > 0$.
A
Both A and R are true and R is the correct explanation of A
B
Both A and R are true but R is NOT the correct explanation of A
C
A is true but R is false
D
A is false but R is true
Solution: A is false (because an internal point cannot generate a chord of contact), but R is true.