120065
The radius of the circle which cuts all the three circles \(\text { circles }\) \(x^2+y^2+4 x-4 y+3=0 x^2+y^2-4 x-4 y+3=0,\) \(x^2+y^2+4 x+4 y+3=0\) orthogonally is
1 1
2 \(\sqrt{3}\)
3 \(\sqrt{5}\)
4 \(\sqrt{7}\)
Explanation:
B Equation of the radical axes are- \(\mathrm{S}_1-\mathrm{S}_2=0\) and \(\mathrm{S}_2-\mathrm{S}_3=0\) Which gives \(-8 \mathrm{x}=0\) \(-8 \mathrm{y}=0\) \(\therefore\) Radical centre \(=(0,0)\) \(\therefore\) Radius \(=\sqrt{\mathrm{S}_1}=\sqrt{\mathrm{S}_2}=\sqrt{\mathrm{S}_3}\) \(=\sqrt{\mathrm{S}_3}=\sqrt{0+0+4 \times 0+4 \times 0+3}=\sqrt{3}\)
TS EAMCET-19.07.2022
Conic Section
120066
Equation of a common tangent to the circle \(x^2+\) \(y^2=4\) and to the ellipse \(2 x^2+25 y^2=50\) is
B \(\text { Let } \mathrm{y}=\mathrm{mx}+\mathrm{c} \text { is common tangent of circle }\) \(\mathrm{x}^2+\mathrm{y}^2=4 \text { and ellipse } \frac{\mathrm{x}^2}{25}+\frac{\mathrm{y}^2}{2}=1\) \(\therefore \quad \mathrm{y}=\mathrm{mx} \pm 2 \sqrt{1+\mathrm{m}^2}\) \(\text { and } \mathrm{y}=\mathrm{mx} \pm \sqrt{25 \mathrm{~m}^2+2} \text { are pair of coincident lines }\) \(\quad 4\left(1+\mathrm{m}^2\right)=25 \mathrm{~m}^2+2\) \(\Rightarrow \quad 21 \mathrm{~m}^2=2 \Rightarrow \mathrm{m}= \pm \sqrt{\frac{2}{21}}\) \(\therefore \text { Equation of tangent is }\) \(\quad \mathrm{y}=\frac{\sqrt{2}}{\sqrt{21}} \mathrm{x} \pm \frac{2 \sqrt{23}}{\sqrt{21}} \Rightarrow \sqrt{2 \mathrm{x}}-\sqrt{21} \mathrm{y}+2 \sqrt{23}=0\) \(\quad \mathrm{y}=\sqrt{\frac{2}{21}} \mathrm{x} \pm 2 \sqrt{\frac{23}{21}}\) \(\quad=\sqrt{2 \mathrm{x}}-\sqrt{21} \mathrm{y}+2 \sqrt{23}=0\) \(\therefore\) Equation of tangent is \(y=\frac{\sqrt{2}}{\sqrt{21}} x \pm \frac{2 \sqrt{23}}{\sqrt{21}} \Rightarrow \sqrt{2 x}-\sqrt{21} y+2 \sqrt{23}=0\) \(y=\sqrt{\frac{2}{21}} x \pm 2 \sqrt{\frac{23}{21}}\) \(=\sqrt{21} x-\sqrt{21} y+2 \sqrt{23}=0\)
TS EAMCET-14.09.2020
Conic Section
120056
The equation \(x^2+y^2+4 x+6 y+13=0\) represents
1 a pair of coincident lines
2 a pair of concurrent straight lines
3 a parabola
4 a point circle
Explanation:
D \(: \text { The equation, } \mathrm{x}^2+\mathrm{y}^2+4 \mathrm{x}+6 \mathrm{y}+13=0\) \(\Rightarrow\left(\mathrm{x}^2+4 \mathrm{x}+4\right)+\left(\mathrm{y}^2+6 \mathrm{y}+9\right)-4-9+13=0\) \(\Rightarrow(\mathrm{x}+2)^2+(\mathrm{y}+3)^2=0\) \(\text { which represent a point circle centered at }(-2,-3)\)which represent a point circle centered at \((-2,-3)\)
120065
The radius of the circle which cuts all the three circles \(\text { circles }\) \(x^2+y^2+4 x-4 y+3=0 x^2+y^2-4 x-4 y+3=0,\) \(x^2+y^2+4 x+4 y+3=0\) orthogonally is
1 1
2 \(\sqrt{3}\)
3 \(\sqrt{5}\)
4 \(\sqrt{7}\)
Explanation:
B Equation of the radical axes are- \(\mathrm{S}_1-\mathrm{S}_2=0\) and \(\mathrm{S}_2-\mathrm{S}_3=0\) Which gives \(-8 \mathrm{x}=0\) \(-8 \mathrm{y}=0\) \(\therefore\) Radical centre \(=(0,0)\) \(\therefore\) Radius \(=\sqrt{\mathrm{S}_1}=\sqrt{\mathrm{S}_2}=\sqrt{\mathrm{S}_3}\) \(=\sqrt{\mathrm{S}_3}=\sqrt{0+0+4 \times 0+4 \times 0+3}=\sqrt{3}\)
TS EAMCET-19.07.2022
Conic Section
120066
Equation of a common tangent to the circle \(x^2+\) \(y^2=4\) and to the ellipse \(2 x^2+25 y^2=50\) is
B \(\text { Let } \mathrm{y}=\mathrm{mx}+\mathrm{c} \text { is common tangent of circle }\) \(\mathrm{x}^2+\mathrm{y}^2=4 \text { and ellipse } \frac{\mathrm{x}^2}{25}+\frac{\mathrm{y}^2}{2}=1\) \(\therefore \quad \mathrm{y}=\mathrm{mx} \pm 2 \sqrt{1+\mathrm{m}^2}\) \(\text { and } \mathrm{y}=\mathrm{mx} \pm \sqrt{25 \mathrm{~m}^2+2} \text { are pair of coincident lines }\) \(\quad 4\left(1+\mathrm{m}^2\right)=25 \mathrm{~m}^2+2\) \(\Rightarrow \quad 21 \mathrm{~m}^2=2 \Rightarrow \mathrm{m}= \pm \sqrt{\frac{2}{21}}\) \(\therefore \text { Equation of tangent is }\) \(\quad \mathrm{y}=\frac{\sqrt{2}}{\sqrt{21}} \mathrm{x} \pm \frac{2 \sqrt{23}}{\sqrt{21}} \Rightarrow \sqrt{2 \mathrm{x}}-\sqrt{21} \mathrm{y}+2 \sqrt{23}=0\) \(\quad \mathrm{y}=\sqrt{\frac{2}{21}} \mathrm{x} \pm 2 \sqrt{\frac{23}{21}}\) \(\quad=\sqrt{2 \mathrm{x}}-\sqrt{21} \mathrm{y}+2 \sqrt{23}=0\) \(\therefore\) Equation of tangent is \(y=\frac{\sqrt{2}}{\sqrt{21}} x \pm \frac{2 \sqrt{23}}{\sqrt{21}} \Rightarrow \sqrt{2 x}-\sqrt{21} y+2 \sqrt{23}=0\) \(y=\sqrt{\frac{2}{21}} x \pm 2 \sqrt{\frac{23}{21}}\) \(=\sqrt{21} x-\sqrt{21} y+2 \sqrt{23}=0\)
TS EAMCET-14.09.2020
Conic Section
120056
The equation \(x^2+y^2+4 x+6 y+13=0\) represents
1 a pair of coincident lines
2 a pair of concurrent straight lines
3 a parabola
4 a point circle
Explanation:
D \(: \text { The equation, } \mathrm{x}^2+\mathrm{y}^2+4 \mathrm{x}+6 \mathrm{y}+13=0\) \(\Rightarrow\left(\mathrm{x}^2+4 \mathrm{x}+4\right)+\left(\mathrm{y}^2+6 \mathrm{y}+9\right)-4-9+13=0\) \(\Rightarrow(\mathrm{x}+2)^2+(\mathrm{y}+3)^2=0\) \(\text { which represent a point circle centered at }(-2,-3)\)which represent a point circle centered at \((-2,-3)\)
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Conic Section
120065
The radius of the circle which cuts all the three circles \(\text { circles }\) \(x^2+y^2+4 x-4 y+3=0 x^2+y^2-4 x-4 y+3=0,\) \(x^2+y^2+4 x+4 y+3=0\) orthogonally is
1 1
2 \(\sqrt{3}\)
3 \(\sqrt{5}\)
4 \(\sqrt{7}\)
Explanation:
B Equation of the radical axes are- \(\mathrm{S}_1-\mathrm{S}_2=0\) and \(\mathrm{S}_2-\mathrm{S}_3=0\) Which gives \(-8 \mathrm{x}=0\) \(-8 \mathrm{y}=0\) \(\therefore\) Radical centre \(=(0,0)\) \(\therefore\) Radius \(=\sqrt{\mathrm{S}_1}=\sqrt{\mathrm{S}_2}=\sqrt{\mathrm{S}_3}\) \(=\sqrt{\mathrm{S}_3}=\sqrt{0+0+4 \times 0+4 \times 0+3}=\sqrt{3}\)
TS EAMCET-19.07.2022
Conic Section
120066
Equation of a common tangent to the circle \(x^2+\) \(y^2=4\) and to the ellipse \(2 x^2+25 y^2=50\) is
B \(\text { Let } \mathrm{y}=\mathrm{mx}+\mathrm{c} \text { is common tangent of circle }\) \(\mathrm{x}^2+\mathrm{y}^2=4 \text { and ellipse } \frac{\mathrm{x}^2}{25}+\frac{\mathrm{y}^2}{2}=1\) \(\therefore \quad \mathrm{y}=\mathrm{mx} \pm 2 \sqrt{1+\mathrm{m}^2}\) \(\text { and } \mathrm{y}=\mathrm{mx} \pm \sqrt{25 \mathrm{~m}^2+2} \text { are pair of coincident lines }\) \(\quad 4\left(1+\mathrm{m}^2\right)=25 \mathrm{~m}^2+2\) \(\Rightarrow \quad 21 \mathrm{~m}^2=2 \Rightarrow \mathrm{m}= \pm \sqrt{\frac{2}{21}}\) \(\therefore \text { Equation of tangent is }\) \(\quad \mathrm{y}=\frac{\sqrt{2}}{\sqrt{21}} \mathrm{x} \pm \frac{2 \sqrt{23}}{\sqrt{21}} \Rightarrow \sqrt{2 \mathrm{x}}-\sqrt{21} \mathrm{y}+2 \sqrt{23}=0\) \(\quad \mathrm{y}=\sqrt{\frac{2}{21}} \mathrm{x} \pm 2 \sqrt{\frac{23}{21}}\) \(\quad=\sqrt{2 \mathrm{x}}-\sqrt{21} \mathrm{y}+2 \sqrt{23}=0\) \(\therefore\) Equation of tangent is \(y=\frac{\sqrt{2}}{\sqrt{21}} x \pm \frac{2 \sqrt{23}}{\sqrt{21}} \Rightarrow \sqrt{2 x}-\sqrt{21} y+2 \sqrt{23}=0\) \(y=\sqrt{\frac{2}{21}} x \pm 2 \sqrt{\frac{23}{21}}\) \(=\sqrt{21} x-\sqrt{21} y+2 \sqrt{23}=0\)
TS EAMCET-14.09.2020
Conic Section
120056
The equation \(x^2+y^2+4 x+6 y+13=0\) represents
1 a pair of coincident lines
2 a pair of concurrent straight lines
3 a parabola
4 a point circle
Explanation:
D \(: \text { The equation, } \mathrm{x}^2+\mathrm{y}^2+4 \mathrm{x}+6 \mathrm{y}+13=0\) \(\Rightarrow\left(\mathrm{x}^2+4 \mathrm{x}+4\right)+\left(\mathrm{y}^2+6 \mathrm{y}+9\right)-4-9+13=0\) \(\Rightarrow(\mathrm{x}+2)^2+(\mathrm{y}+3)^2=0\) \(\text { which represent a point circle centered at }(-2,-3)\)which represent a point circle centered at \((-2,-3)\)