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CAT Length of the Tangent - Practice Questions & MCQ

Edited By admin | Updated on Oct 04, 2023 04:20 PM | #CAT

Quick Facts

5 Questions around this concept.

Solve by difficulty

The length of the tangent drawn from a point, whose distance from the center of a circle is 5 cm and the radius of the circle is 3 cm is:

View Solution

The length of a tangent from a point (4, 4) to the circle $x^2+y^2=4$ is:

$2\sqrt{14}$

$2\sqrt7$

View Solution

Concepts Covered - 1

Length of the Tangent

PT is a tangent to a circle and CT is radius of circle. Since, PT is perpendicular to CT. So, ΔPTC is a right angled triangle.

Using the pythagorean theorem

$\\\mathrm{\;\;\;\;\;\;PC^2=PT^2+CT^2 } \\\Rightarrow \mathrm{\;PT^2=PC^2-CT^2}\\\text{Length of tangent is PT}\\\text{So,}\\\Rightarrow \;\;\mathrm{PT=\sqrt{PC^2-CT^2}}$

Now, if we have given equation of circle $\mathrm{ x^2+y^2=a^2}$ and coordinate of point P(x₁, y₁), then

$\\\text {In } \Delta \mathrm{PTC}, \quad \mathrm{PT}^{2}=\mathrm{PC}^{2}-\mathrm{CT}^{2}\\ \text{Here for the circle} \mathrm{x^{2}+y^{2}=a^{2}}, \text{ coordinate of C is }(0,0)\\ \text { Hence, } \mathrm{PT}^{2}=(\sqrt{\left(\mathrm{x}_{1}-0\right)^{2}+\left(\mathrm{y}_{1}-0\right)^{2}}) ^{2}-(\sqrt{\mathrm{a}^{2}})^{2} \\ \Rightarrow \quad \mathrm{PT}=\sqrt{\mathrm{x}_{1}^{2}+\mathrm{y}_{1}^{2}-\mathrm{a}^{2}} \quad[\because \mathrm{CT}=\mathrm{radius}=\mathrm{a}]$

If we have given general equation of circle $\mathrm{ x^2+y^2+2gx+2fy+c=0}$ and coordinate of point P(x₁, y₁) from where tanget is drawn to the circle then, the length of tangent will be $\sqrt{\mathrm{ x_1^2+y_1^2+2gx_1+2fy_1+c=0}}$ .

Study other Related Concepts

CAT Length of the Tangent Current Topic

CAT Basic Terms and Definitions of Lines and Angles

CAT Various Types of Angle

CAT Linear Pair of Angles

CAT Vertically Opposite Angles

CAT Corresponding Angle Axioms

CAT Alternate Interior Angle Theorem

CAT Interior Angle Theorem

CAT Parallel and a Transversal Lines

CAT Introduction to Triangles

CAT Congruence of Triangles

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