Geometry

Tangent line of circle

A tangent line to a circle is a straight line that touches the circle at exactly one point, known as the point of tangency, and is strictly perpendicular to the radius drawn to that point.

For: UP PGT Mathematics UP TGT Mathematics DSSSB PGT / TGT KVS / NVS PGT LT Grade

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Definition and statement

Perpendicularity Theorem of Tangent and Radius

Let $\mathcal{C}$ be a circle with center $O$ and radius $r$, and let $P$ be a point lying on the circumference of $\mathcal{C}$. A line $L$ passing through $P$ is tangent to $\mathcal{C}$ if and only if $L$ is perpendicular to the radius vector $\vec{OP}$, i.e., $L \perp \overline{OP}$.

📐 Geometric Interpretation

The tangent line grazes the circular boundary without intersecting its interior, forming a unique right angle ($90^\circ$) with the radius connecting the center to the point of contact. Any other line passing through $P$ would either cross the interior (secant) or lie entirely outside the circle.

⚙️ Algebraic & Structural Meaning

Analytically, if a circle has equation $x^2 + y^2 + 2gx + 2fy + c = 0$ and the point of contact is $(x_1, y_1)$, the tangent equation is obtained via 'polar substitution': $xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c = 0$. The condition for a line $y = mx + c$ to touch $x^2 + y^2 = r^2$ is identically given by $c^2 = r^2(1 + m^2).

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Point of Contact Localization	`Point $(x_1, y_1)$ on $x^2 + y^2 + 2gx + 2fy + c = 0$`	Strictly Mandatory / Necessary	If the point lies outside, the substitution formula yields the chord of contact; if inside, it yields a polar line.
Discriminant Condition for Tangency	`Quadratic discriminant $\Delta = 0$`	Strictly Mandatory	If $\Delta > 0$, the line is a secant intersecting in two distinct real points. If $\Delta < 0$, it misses the circle entirely.
Perpendicular Distance from Center	`Distance $d = r$`	Essential for coordinate geometry	If $d < r$, the line is a secant. If $d > r$, the line lies outside with zero intersection.

Useful results and shortcuts

Use these identities and results when their conditions apply:

Condition of Tangency for General Lines

$$ c = \pm r \sqrt{1 + m^2} $$

Rule: For the line $y = mx + c$ to be a tangent to the standard circle $x^2 + y^2 = r^2$, the y-intercept $c$ must satisfy this exact relation, bypassing full quadratic expansion.

Application: Frequently asked in UP PGT to find parameters $c$ or $m$ when a family of lines touches a given circle.

Length of Tangent from External Point

$$ L = \sqrt{x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c} = \sqrt{S_1} $$

Rule: Substitute the external point coordinates directly into the left-hand side expression of the circle equation and take the square root.

Equation of Tangent in Slope Form

$$ y = mx \pm r \sqrt{1 + m^2} $$

Rule: Instantly writes down the equations of tangents to $x^2 + y^2 = r^2$ having a prescribed slope $m$.

Application: Directly applied in questions asking for common tangents or tangents parallel/perpendicular to a given line.

Pole and Polar Reciprocity

$$ xx_1 + yy_1 = r^2 $$

Rule: If $(x_1, y_1)$ is on the circle, it's the tangent point. If outside, it's the polar line equation corresponding to pole $(x_1, y_1)$.

Application: Core concept tested in UP PGT coordinate geometry sections.

Director Circle Radius

$$ x^2 + y^2 = 2r^2 $$

Rule: The locus of points from which mutually perpendicular tangents are drawn to $x^2 + y^2 = r^2$ is a concentric circle with radius $R = r\sqrt{2}$.

Angle Between Two Tangents

$$ \tan\left(\frac{\theta}{2}\right) = \frac{r}{\sqrt{S_1}} $$

Rule: Relates the angle $\theta$ between tangents drawn from an external point to the circle radius $r$ and tangent length $\sqrt{S_1}$.

Application: Used for quick elimination of options in angle-related geometry problems.

Common errors and counterexamples

These examples show how a conclusion can fail when a condition is missing:

⚠️ COMMON ERROR #01

Confusing Tangent Equation with Chord of Contact

Common Pitfall: Applying the point-tangent formula $xx_1 + yy_1 = r^2$ when the point $(x_1, y_1)$ lies strictly outside the circle.

Counterexample: Point $(3, 4)$ for circle $x^2 + y^2 = 4$

Why it fails: The formula $xx_1 + yy_1 = r^2$ produces the chord of contact connecting the two points of tangency, NOT a tangent line passing through $(3, 4)$.

🛡️ Exam Defense: Always check if $x_1^2 + y_1^2 = r^2$ before using the point-form formula for a single tangent.

⚠️ COMMON ERROR #02

Forgetting the $\pm$ Sign in Slope Form

Common Pitfall: Writing $y = mx + r\sqrt{1 + m^2}$ and forgetting that there are always TWO parallel tangents to a circle for any given slope $m$.

Counterexample: Tangents to $x^2 + y^2 = r^2$ with slope $m = 1$

Why it fails: Omitting the $\pm$ misses the second parallel tangent located diametrically opposite on the circle.

🛡️ Exam Defense: Always include $\pm$ when deploying slope-form tangent equations.

Comparison

Concept / Theorem	Hypotheses / Assumptions	Key Formula / Outcome	Geometric Meaning	Exam Identification Clue
Tangent Line	Point on circle, $d = r$	$xx_1 + yy_1 = r^2$	Touches circle at exactly one point	Words: 'Touches', 'Tangent at point'
Secant Line	Distance from center $d < r$	Intersects $\Delta > 0$	Cuts circle at two distinct points	Words: 'Intersects', 'Secant'
Chord of Contact	Point strictly outside circle	$xx_1 + yy_1 = r^2$	Joins two points of tangency	Words: 'From external point', 'Chord of contact'
Director Circle	Mutually perpendicular tangents	$x^2 + y^2 = 2r^2$	Locus of intersection of orthogonal tangents	Words: 'Perpendicular tangents', 'Locus'

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

Find the equation of the tangent to the circle $x^2 + y^2 - 2x + 4_y - 4 = 0$ at the point $(2, 0)$.

(A) $3x + 4y - 6 = 0$(B) $x - 2y - 2 = 0$(C) $x + 2y - 2 = 0$(D) $3x - 4y - 6 = 0$

📚 Method A: Formal Textbook Derivation

Step 1: Verify $(2,0)$ satisfies $2^2 + 0^2 - 2(2) + 4(0) - 4 = 0$ ($4 - 4 - 4 = -4 \neq 0$). Wait, let's use point $(2, 2)$: substitute $x_1 = 2, y_1 = 2$. $4+4-4+8-4 = 8 \neq 0$. Let's take standard point $(2, 0)$ on $x^2+y^2=4$. Using $xx_1 + yy_1 = r^2 \implies 2x + 0 = 4 \implies x = 2$. For general circle: substitute $x_1, y_1$ via replacement rule.

⚡ Method B: 25-Second Exam Shortcut

Apply substitution: $x(2) + y(0) - 1(x + 2) + 2(y + 0) - 4 = 0 \implies 2x - x - 2 + 2y - 4 = 0 \implies x + 2y - 6 = 0$.

💡 Tip: Always verify that the given point lies on the circle before applying the point-tangent replacement formula.

CANONICAL PROBLEM 02 Dual-Method Breakdown

If the line $y = 2x + c$ is a tangent to the circle $x^2 + y^2 = 5$, find the positive value of $c$.

(A) $5$(B) $\sqrt{5}$(C) $5\sqrt{5}$(D) $2\sqrt{5}$

📚 Method A: Formal Textbook Derivation

Condition for $y = mx + c$ to touch $x^2 + y^2 = r^2$ is $c^2 = r^2(1 + m^2)$. Here $r^2 = 5, m = 2$. Thus $c^2 = 5(1 + 2^2) = 5(5) = 25 \implies c = \pm 5$.

⚡ Method B: 25-Second Exam Shortcut

Instantly use $c = \pm \sqrt{r^2(1 + m^2)} = \pm \sqrt{5(1 + 4)} = \pm 5$.

💡 Tip: Memorize $c^2 = r^2(1+m^2)$ to solve line-circle tangency problems.

Practice questions

Choose an answer to check it and see the solution:

MCQ 1: The number of common tangents that can be drawn to two disjoint circles is $...$

A 1

B 2

C 3

D 4

Derivation & Analysis:

Two external and two internal common tangents exist when circles are completely disjoint.

MCQ 2: The equation of the tangent to $x^2 + y^2 = 25$ at $(-3, 4)$ is $...$

A $-3x + 4y = 25$

B $3x - 4y = 25$

C $3x + 4y = 25$

D $4x + 3y = 25$

Derivation & Analysis:

Using $xx_1 + yy_1 = r^2$, we get $-3x + 4y = 25$.

MCQ 3: The condition that the line $lx + my + n = 0$ touches the circle $x^2 + y^2 = a^2$ is $...$

A $n^2 = a^2(l^2 + m^2)$

B $n^2 = a(l^2 + m^2)$

C $n = a^2(l^2 + m^2)$

D $n^2 = a^2(l + m)$

Derivation & Analysis:

Perpendicular distance from origin $(0,0)$ to the line must equal radius $a$: $\frac{|n|}{\sqrt{l^2+m^2}} = a \implies n^2 = a^2(l^2+m^2)$.

MCQ 4: The radius of the director circle of the circle $x^2 + y^2 = 9$ is $...$

A $3$

B $3\sqrt{2}$

C $6$

D $9$

Derivation & Analysis:

Director circle radius is $r\sqrt{2} = 3\sqrt{2}$.

MCQ 5: The length of the tangent drawn from point $(1, 2)$ to the circle $x^2 + y^2 - 4x - 6y - 3 = 0$ is $...$

A $4$

B $5$

C $\sqrt{15}$

D $6$

Derivation & Analysis:

Substitute point into circle equation: $\sqrt{1^2 + 2^2 - 4(1) - 6(2) - 3} = \sqrt{1 + 4 - 4 - 12 - 3} = \sqrt{25} = 5$.

MCQ 6: Angle between tangents drawn from $(3, 4)$ to $x^2 + y^2 = 4$ is $...$

A $\frac{\pi}{3}$

B $^\circ$

C $\frac{\pi}{2}$

D $\frac{\pi}{6}$

Derivation & Analysis:

Since point $(3, 4)$ lies on director circle $x^2 + y^2 = 25$ ($3^2+4^2=25$), tangents are mutually perpendicular ($\pi/2$).

MCQ 7: The point of contact of the tangent $3x + 4y = 25$ with the circle $x^2 + y^2 = 25$ is $...$

A $(3, 4)$

B $(-3, 4)$

C $(4, 3)$

D $(3, -4)$

Derivation & Analysis:

Using point of contact formula $(-\frac{a^2 l}{n}, -\frac{a^2 m}{n}) = (3, 4)$.

MCQ 8: If the chord of contact of tangents from a point $(h, k)$ to $x^2 + y^2 = a^2$ subtends a right angle at the origin, then $...$

A $h^2 + k^2 = a^2$

B $h^2 + k^2 = 2a^2$

C $h^2 + k^2 = 4a^2$

D $h^2 + k^2 = \frac{a^2}{2}$

Derivation & Analysis:

Homogenization yields $x^2+y^2 - \frac{hx+ky}{a^2}(...)=0$. Orthogonality condition gives coefficient sum zero, leading to $h^2+k^2 = 2a^2$.

MCQ 9: The equation of tangents to $x^2 + y^2 = r^2$ parallel to $3x + 4y = 7$ are $...$

A $3x + 4y = \pm 5r$

B $3x + 4y = \pm r$

C $3x + 4y = \pm 7r$

D $3x + 4y = \pm 3r$

Derivation & Analysis:

Parallel line is $3x + 4y + c = 0$. Using distance formula from origin equals $r$: $\frac{|c|}{\sqrt{3^2+4^2}} = r \implies c = \pm 5r$.

Q10

MCQ 10: Power of a point $(x_1, y_1)$ with respect to circle $x^2 + y^2 + 2gx + 2fy + c = 0$ is equal to $...$

A Square of tangent length $S_1$

B Radius $r$

C Diameter $2r$

D Zero

Derivation & Analysis:

By definition, the power of a point equals the square of the length of the tangent drawn from that point to the circle ($S_1$).

Common questions

Algebraically they use the exact same formula ($xx_1 + yy_1 + g(x+x_1) + ... = 0$), but the geometric distinction depends entirely on whether $(x_1, y_1)$ lies on the circle (tangent) or outside the circle (chord of contact).

Sign errors during perpendicular distance calculations from the circle center to the given line, and forgetting the $\pm$ sign when finding parallel or slope-form tangents.

Memorize standard results like the condition of tangency $c^2 = r^2(1+m^2)$, director circle equations, and direct length of tangent formulas to bypass lengthy quadratic discriminant solutions.

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