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MCQEasyJEE 2023Circle Equation & Properties

JEE Mathematics 2023 Question with Solution

The number of common tangents to the circles x2+y2−18x−15y+131=0x^2+y^2-18x-15y+131=0 and x2+y2−6x−6y−7=0x^2+y^2-6x-6y-7=0 is

  • A

    11

  • B

    22

  • C

    33

  • D

    44

Answer

Correct answer:C

Step-by-step solution

Standard Method

Given: The circles are x2+y2−18x−15y+131=0x^2+y^2-18x-15y+131=0 and x2+y2−6x−6y−7=0x^2+y^2-6x-6y-7=0.

Find: The number of common tangents to these two circles.

Write each circle in standard form.

(x−9)2+(y−152)2=254(x-9)^2+\left(y-\tfrac{15}{2}\right)^2=\tfrac{25}{4}

So, the first circle has center C1(9,152)C_1\left(9,\tfrac{15}{2}\right) and radius r1=52r_1=\tfrac{5}{2}.

For the second circle,

(x−3)2+(y−3)2=25(x-3)^2+(y-3)^2=25

So, the second circle has center C2(3,3)C_2(3,3) and radius r2=5r_2=5.

Now find the distance between the centers.

d=(9−3)2+(152−3)2d=\sqrt{(9-3)^2+\left(\tfrac{15}{2}-3\right)^2} =36+(92)2=\sqrt{36+\left(\tfrac{9}{2}\right)^2} =2254=152=\sqrt{\tfrac{225}{4}}=\tfrac{15}{2}

Compare Distance and Radii

Compare dd with r1+r2r_1+r_2 and ∣r1−r2∣|r_1-r_2|.

r1+r2=52+5=152r_1+r_2=\tfrac{5}{2}+5=\tfrac{15}{2} ∣r1−r2∣=5−52=52|r_1-r_2|=5-\tfrac{5}{2}=\tfrac{5}{2}

Here,

d=r1+r2d=r_1+r_2

So the circles touch each other externally.

When two circles touch externally, they have:

  • two direct common tangents
  • one transverse common tangent

Hence, the total number of common tangents is 33.

Therefore, the correct option is C.

Quick Tip: Always compare the distance between centers with the sum and difference of radii to quickly determine the number of common tangents between two circles.

Common mistakes

  • A common mistake is computing the centers incorrectly while completing the square. This gives wrong radii and wrong distance comparison. Rewrite each circle carefully in standard form before comparing their positions.

  • Some students compare only dd and r1+r2r_1+r_2 but ignore the geometric meaning. Here d=r1+r2d=r_1+r_2 means the circles touch externally, which leads to exactly three common tangents.

  • Another mistake is assuming touching circles always have two common tangents. That is true for internal touching, not external touching. For external touching, there are two direct tangents and one transverse tangent.

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