119895
The angle between the two tangents drawn from origin to the circle \(x^2+y^2-14 x+2 y+25\) \(=\mathbf{0}\)
1 \(0^{\circ}\)
2 \(45^{\circ}\)
3 \(90^{\circ}\)
4 \(60^{\circ}\)
Explanation:
C : If \(2 \alpha\) is the angle between the two tangents from the origin to the circle. \(\tan \alpha=\frac{\mathrm{CT}_1}{\mathrm{OT}_1}=\frac{\sqrt{49+1-25}}{\sqrt{25}}=1\) \(\alpha=\tan ^{-1}(1)=45^{\circ}\) \(2 \alpha=2 \times 45^{\circ}=90^{\circ}=\frac{\pi}{2}\)Angle between the tangents is \(\frac{\pi}{2}\)
AP EAMCET-23.08.2021
Conic Section
119896
The line \(a x+b y+c=0\) is normal to the circle \(\mathbf{x}^2+\mathbf{y}^2+\mathbf{2 g x}+\mathbf{2 f y}+\mathbf{d}=\mathbf{0}\) if \(\qquad\)
1 \(a g+b f+c=0\)
2 \(\overline{a g+b} f-c=0\)
3 \(\mathrm{ag}-\mathrm{bf}+\mathrm{c}=0\)
4 ag \(-\mathrm{bf}-\mathrm{c}=0\)
Explanation:
B The centre of given circle is \((-\mathrm{g},-\mathrm{f})\). If the given line \(\mathrm{ax}+\mathrm{bx}+\mathrm{c}=0\) is normal to the circle, then it passes through the centre of circle \(\therefore \mathrm{a}(-\mathrm{g})+\mathrm{b}(-\mathrm{f})+\mathrm{c}=0\) \(\mathrm{ag}+\mathrm{bf}-\mathrm{c}=0\)
AP EAMCET-23.08.2021
Conic Section
119897
The shortest distance from the line \(3 x+4 y=25\) to the circle \(x^2+y^2-6 x+8 y=0\) is
1 \(\frac{9}{5}\)
2 \(\frac{7}{5}\)
3 \(\frac{8}{5}\)
4 \(\frac{13}{5}\)
Explanation:
B Given, equation of circle is \(\mathrm{x}^2+\mathrm{y}^2-6 \mathrm{x}+8 \mathrm{y}=0\) \((x-3)^2+(y+4)^2=25\) shortest distance \((\mathrm{d})=\) distance of line from centre - Radius \(=\frac{32}{5}-5=\frac{32-25}{5} \mathrm{~d}=\frac{7}{5}\)
AP EAMCET-08.07.2022
Conic Section
119898
A circle touches the Y-axis at the point \((0,4)\) and passes through the point \((2,0)\). Which of the following lines is not a tangent to this circle?
1 \(4 x-3 y+17=0\)
2 \(3 x+4 y-6=0\)
3 \(4 x+3 y-8=0\)
4 \(3 x-4 y-24=0\)
Explanation:
C Given point \((0,4)\) Equation of family of circle \((\mathrm{x}-0)^2+(\mathrm{y}-4)^2+\lambda \mathrm{x}=0\) passed through \((2,0)\) \(\therefore(2-0)^2+(0-4)^2+2 \lambda=0\) \(2^2+(-4)^2+2 \lambda=0\) \(4+16+2 \lambda=0\) \(\lambda=-10\) Equation of circle \(\mathrm{x}^2+\mathrm{y}^2+16-8 \mathrm{y}+(-10 \mathrm{x})=0\) \(\mathrm{x}^2+\mathrm{y}^2-10 \mathrm{x}-8 \mathrm{y}+16=0\) centre, \((+5,4)\) Radius \(=\sqrt{25+16-16}=\sqrt{25}=5\) \(\Rightarrow\) checking option \(\mathrm{A}\) tangent \(4 x+3 y-8=0\) distance from centre \(=5\) \(\left|\frac{(4 \times 5)+(3 \times 4)-8}{\sqrt{(4)^2+(3)^2}}\right|=\left|\frac{20+12-8}{5}\right|=\left|\frac{24}{5}\right| \neq 5\) \(\therefore 4 \mathrm{x}+3 \mathrm{y}-8=0\) not a tangent to circle.
119895
The angle between the two tangents drawn from origin to the circle \(x^2+y^2-14 x+2 y+25\) \(=\mathbf{0}\)
1 \(0^{\circ}\)
2 \(45^{\circ}\)
3 \(90^{\circ}\)
4 \(60^{\circ}\)
Explanation:
C : If \(2 \alpha\) is the angle between the two tangents from the origin to the circle. \(\tan \alpha=\frac{\mathrm{CT}_1}{\mathrm{OT}_1}=\frac{\sqrt{49+1-25}}{\sqrt{25}}=1\) \(\alpha=\tan ^{-1}(1)=45^{\circ}\) \(2 \alpha=2 \times 45^{\circ}=90^{\circ}=\frac{\pi}{2}\)Angle between the tangents is \(\frac{\pi}{2}\)
AP EAMCET-23.08.2021
Conic Section
119896
The line \(a x+b y+c=0\) is normal to the circle \(\mathbf{x}^2+\mathbf{y}^2+\mathbf{2 g x}+\mathbf{2 f y}+\mathbf{d}=\mathbf{0}\) if \(\qquad\)
1 \(a g+b f+c=0\)
2 \(\overline{a g+b} f-c=0\)
3 \(\mathrm{ag}-\mathrm{bf}+\mathrm{c}=0\)
4 ag \(-\mathrm{bf}-\mathrm{c}=0\)
Explanation:
B The centre of given circle is \((-\mathrm{g},-\mathrm{f})\). If the given line \(\mathrm{ax}+\mathrm{bx}+\mathrm{c}=0\) is normal to the circle, then it passes through the centre of circle \(\therefore \mathrm{a}(-\mathrm{g})+\mathrm{b}(-\mathrm{f})+\mathrm{c}=0\) \(\mathrm{ag}+\mathrm{bf}-\mathrm{c}=0\)
AP EAMCET-23.08.2021
Conic Section
119897
The shortest distance from the line \(3 x+4 y=25\) to the circle \(x^2+y^2-6 x+8 y=0\) is
1 \(\frac{9}{5}\)
2 \(\frac{7}{5}\)
3 \(\frac{8}{5}\)
4 \(\frac{13}{5}\)
Explanation:
B Given, equation of circle is \(\mathrm{x}^2+\mathrm{y}^2-6 \mathrm{x}+8 \mathrm{y}=0\) \((x-3)^2+(y+4)^2=25\) shortest distance \((\mathrm{d})=\) distance of line from centre - Radius \(=\frac{32}{5}-5=\frac{32-25}{5} \mathrm{~d}=\frac{7}{5}\)
AP EAMCET-08.07.2022
Conic Section
119898
A circle touches the Y-axis at the point \((0,4)\) and passes through the point \((2,0)\). Which of the following lines is not a tangent to this circle?
1 \(4 x-3 y+17=0\)
2 \(3 x+4 y-6=0\)
3 \(4 x+3 y-8=0\)
4 \(3 x-4 y-24=0\)
Explanation:
C Given point \((0,4)\) Equation of family of circle \((\mathrm{x}-0)^2+(\mathrm{y}-4)^2+\lambda \mathrm{x}=0\) passed through \((2,0)\) \(\therefore(2-0)^2+(0-4)^2+2 \lambda=0\) \(2^2+(-4)^2+2 \lambda=0\) \(4+16+2 \lambda=0\) \(\lambda=-10\) Equation of circle \(\mathrm{x}^2+\mathrm{y}^2+16-8 \mathrm{y}+(-10 \mathrm{x})=0\) \(\mathrm{x}^2+\mathrm{y}^2-10 \mathrm{x}-8 \mathrm{y}+16=0\) centre, \((+5,4)\) Radius \(=\sqrt{25+16-16}=\sqrt{25}=5\) \(\Rightarrow\) checking option \(\mathrm{A}\) tangent \(4 x+3 y-8=0\) distance from centre \(=5\) \(\left|\frac{(4 \times 5)+(3 \times 4)-8}{\sqrt{(4)^2+(3)^2}}\right|=\left|\frac{20+12-8}{5}\right|=\left|\frac{24}{5}\right| \neq 5\) \(\therefore 4 \mathrm{x}+3 \mathrm{y}-8=0\) not a tangent to circle.
119895
The angle between the two tangents drawn from origin to the circle \(x^2+y^2-14 x+2 y+25\) \(=\mathbf{0}\)
1 \(0^{\circ}\)
2 \(45^{\circ}\)
3 \(90^{\circ}\)
4 \(60^{\circ}\)
Explanation:
C : If \(2 \alpha\) is the angle between the two tangents from the origin to the circle. \(\tan \alpha=\frac{\mathrm{CT}_1}{\mathrm{OT}_1}=\frac{\sqrt{49+1-25}}{\sqrt{25}}=1\) \(\alpha=\tan ^{-1}(1)=45^{\circ}\) \(2 \alpha=2 \times 45^{\circ}=90^{\circ}=\frac{\pi}{2}\)Angle between the tangents is \(\frac{\pi}{2}\)
AP EAMCET-23.08.2021
Conic Section
119896
The line \(a x+b y+c=0\) is normal to the circle \(\mathbf{x}^2+\mathbf{y}^2+\mathbf{2 g x}+\mathbf{2 f y}+\mathbf{d}=\mathbf{0}\) if \(\qquad\)
1 \(a g+b f+c=0\)
2 \(\overline{a g+b} f-c=0\)
3 \(\mathrm{ag}-\mathrm{bf}+\mathrm{c}=0\)
4 ag \(-\mathrm{bf}-\mathrm{c}=0\)
Explanation:
B The centre of given circle is \((-\mathrm{g},-\mathrm{f})\). If the given line \(\mathrm{ax}+\mathrm{bx}+\mathrm{c}=0\) is normal to the circle, then it passes through the centre of circle \(\therefore \mathrm{a}(-\mathrm{g})+\mathrm{b}(-\mathrm{f})+\mathrm{c}=0\) \(\mathrm{ag}+\mathrm{bf}-\mathrm{c}=0\)
AP EAMCET-23.08.2021
Conic Section
119897
The shortest distance from the line \(3 x+4 y=25\) to the circle \(x^2+y^2-6 x+8 y=0\) is
1 \(\frac{9}{5}\)
2 \(\frac{7}{5}\)
3 \(\frac{8}{5}\)
4 \(\frac{13}{5}\)
Explanation:
B Given, equation of circle is \(\mathrm{x}^2+\mathrm{y}^2-6 \mathrm{x}+8 \mathrm{y}=0\) \((x-3)^2+(y+4)^2=25\) shortest distance \((\mathrm{d})=\) distance of line from centre - Radius \(=\frac{32}{5}-5=\frac{32-25}{5} \mathrm{~d}=\frac{7}{5}\)
AP EAMCET-08.07.2022
Conic Section
119898
A circle touches the Y-axis at the point \((0,4)\) and passes through the point \((2,0)\). Which of the following lines is not a tangent to this circle?
1 \(4 x-3 y+17=0\)
2 \(3 x+4 y-6=0\)
3 \(4 x+3 y-8=0\)
4 \(3 x-4 y-24=0\)
Explanation:
C Given point \((0,4)\) Equation of family of circle \((\mathrm{x}-0)^2+(\mathrm{y}-4)^2+\lambda \mathrm{x}=0\) passed through \((2,0)\) \(\therefore(2-0)^2+(0-4)^2+2 \lambda=0\) \(2^2+(-4)^2+2 \lambda=0\) \(4+16+2 \lambda=0\) \(\lambda=-10\) Equation of circle \(\mathrm{x}^2+\mathrm{y}^2+16-8 \mathrm{y}+(-10 \mathrm{x})=0\) \(\mathrm{x}^2+\mathrm{y}^2-10 \mathrm{x}-8 \mathrm{y}+16=0\) centre, \((+5,4)\) Radius \(=\sqrt{25+16-16}=\sqrt{25}=5\) \(\Rightarrow\) checking option \(\mathrm{A}\) tangent \(4 x+3 y-8=0\) distance from centre \(=5\) \(\left|\frac{(4 \times 5)+(3 \times 4)-8}{\sqrt{(4)^2+(3)^2}}\right|=\left|\frac{20+12-8}{5}\right|=\left|\frac{24}{5}\right| \neq 5\) \(\therefore 4 \mathrm{x}+3 \mathrm{y}-8=0\) not a tangent to circle.
NEET Test Series from KOTA - 10 Papers In MS WORD
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Conic Section
119895
The angle between the two tangents drawn from origin to the circle \(x^2+y^2-14 x+2 y+25\) \(=\mathbf{0}\)
1 \(0^{\circ}\)
2 \(45^{\circ}\)
3 \(90^{\circ}\)
4 \(60^{\circ}\)
Explanation:
C : If \(2 \alpha\) is the angle between the two tangents from the origin to the circle. \(\tan \alpha=\frac{\mathrm{CT}_1}{\mathrm{OT}_1}=\frac{\sqrt{49+1-25}}{\sqrt{25}}=1\) \(\alpha=\tan ^{-1}(1)=45^{\circ}\) \(2 \alpha=2 \times 45^{\circ}=90^{\circ}=\frac{\pi}{2}\)Angle between the tangents is \(\frac{\pi}{2}\)
AP EAMCET-23.08.2021
Conic Section
119896
The line \(a x+b y+c=0\) is normal to the circle \(\mathbf{x}^2+\mathbf{y}^2+\mathbf{2 g x}+\mathbf{2 f y}+\mathbf{d}=\mathbf{0}\) if \(\qquad\)
1 \(a g+b f+c=0\)
2 \(\overline{a g+b} f-c=0\)
3 \(\mathrm{ag}-\mathrm{bf}+\mathrm{c}=0\)
4 ag \(-\mathrm{bf}-\mathrm{c}=0\)
Explanation:
B The centre of given circle is \((-\mathrm{g},-\mathrm{f})\). If the given line \(\mathrm{ax}+\mathrm{bx}+\mathrm{c}=0\) is normal to the circle, then it passes through the centre of circle \(\therefore \mathrm{a}(-\mathrm{g})+\mathrm{b}(-\mathrm{f})+\mathrm{c}=0\) \(\mathrm{ag}+\mathrm{bf}-\mathrm{c}=0\)
AP EAMCET-23.08.2021
Conic Section
119897
The shortest distance from the line \(3 x+4 y=25\) to the circle \(x^2+y^2-6 x+8 y=0\) is
1 \(\frac{9}{5}\)
2 \(\frac{7}{5}\)
3 \(\frac{8}{5}\)
4 \(\frac{13}{5}\)
Explanation:
B Given, equation of circle is \(\mathrm{x}^2+\mathrm{y}^2-6 \mathrm{x}+8 \mathrm{y}=0\) \((x-3)^2+(y+4)^2=25\) shortest distance \((\mathrm{d})=\) distance of line from centre - Radius \(=\frac{32}{5}-5=\frac{32-25}{5} \mathrm{~d}=\frac{7}{5}\)
AP EAMCET-08.07.2022
Conic Section
119898
A circle touches the Y-axis at the point \((0,4)\) and passes through the point \((2,0)\). Which of the following lines is not a tangent to this circle?
1 \(4 x-3 y+17=0\)
2 \(3 x+4 y-6=0\)
3 \(4 x+3 y-8=0\)
4 \(3 x-4 y-24=0\)
Explanation:
C Given point \((0,4)\) Equation of family of circle \((\mathrm{x}-0)^2+(\mathrm{y}-4)^2+\lambda \mathrm{x}=0\) passed through \((2,0)\) \(\therefore(2-0)^2+(0-4)^2+2 \lambda=0\) \(2^2+(-4)^2+2 \lambda=0\) \(4+16+2 \lambda=0\) \(\lambda=-10\) Equation of circle \(\mathrm{x}^2+\mathrm{y}^2+16-8 \mathrm{y}+(-10 \mathrm{x})=0\) \(\mathrm{x}^2+\mathrm{y}^2-10 \mathrm{x}-8 \mathrm{y}+16=0\) centre, \((+5,4)\) Radius \(=\sqrt{25+16-16}=\sqrt{25}=5\) \(\Rightarrow\) checking option \(\mathrm{A}\) tangent \(4 x+3 y-8=0\) distance from centre \(=5\) \(\left|\frac{(4 \times 5)+(3 \times 4)-8}{\sqrt{(4)^2+(3)^2}}\right|=\left|\frac{20+12-8}{5}\right|=\left|\frac{24}{5}\right| \neq 5\) \(\therefore 4 \mathrm{x}+3 \mathrm{y}-8=0\) not a tangent to circle.
