NIMCET 2021 — Mathematics PYQ

Analysis of Circles 

1. The first circle's equation is given as $x^2+y^2=4$. 
• The center of this circle, $C_1$, is determined to be $(0,0)$. 
• The radius of this circle, $R_1$, is found by taking the square root of the constant term, resulting in $R_1=\sqrt{4}=2$. 
2. The second circle's equation is given as $x^2+y^2-6x-8y=24$.     
• This equation is rewritten in standard form by completing the square: $(x^2-6x+9)+(y^2-8y+16)=24+9+16$. 
• This simplifies to $(x-3{)}^2+(y-4{)}^2=49$. 
• The center of this circle, $C_2$, is determined to be $(3,4)$. 
• The radius of this circle, $R_2$, is found by taking the square root of the constant term, resulting in $R_2=\sqrt{49}=7$. 
Distance Between Centers 
1. The distance between the centers $C_1(0,0)$ and $C_2(3,4)$ is calculated using the distance formula: $d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$. 
2. Substituting the coordinates, $d=\sqrt{(3-0{)}^2+(4-0{)}^2}=\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}$. 
3. Therefore, the distance between the centers is $d=5$. 
Relationship Between Circles 
1. The difference between the radii is calculated: $R_2-R_1=7-2=5$. 
2. It is observed that the distance between the centers, $d=5$, is equal to the difference between the radii, $R_2-R_1=5$. 
3. This condition, $d=R_2-R_1$, indicates that the two circles touch each other internally. 
Number of Common Tangents 
1. When two circles touch each other internally, there is only one common tangent. This tangent exists at the point of contact between the two circles. 
Final Answer 
The number of common tangents to the given circles is $1$.