120793
If the circle \(x^2+y^2=a^2\) intersects the hyperbola \(x y=c^2\) in four points \(\left(x_i, y_i\right)\) for \(i=1,2,3\) and 4 , then \(\mathbf{y}_1+\mathbf{y}_2+\mathbf{y}_3+\mathbf{y}_4\) equals
1 0
2 \(\mathrm{c}\)
3 a
4 \(\mathrm{c}^4\)
Explanation:
A Given, circle : \(x^2+y^2=a^2\)
\(H: x y=c^2\)
\(\therefore \quad \mathrm{x}^2+\frac{\mathrm{c}^4}{\mathrm{x}^2}=\mathrm{a}^2\)
\(x^4-a^2 x^2+c^4=0\)
\(\mathrm{x}_1+\mathrm{x}_2+\mathrm{x}_3+\mathrm{x}_4=0\)
\(\text { or } \mathrm{y}_1+\mathrm{y}_2+\mathrm{y}_3+\mathrm{y}_4=0\)
Now, sum of roots:
AP EAMCET-2009
Hyperbola
120794
If \(x=9\) is a chord of contact of the hyperbola \(x^2\) \(-y^2=9\), then the equation of the tangent at one of the points of contact is
1 \(x+\sqrt{3 y}+2=0\)
2 \(3 \mathrm{x}-2 \sqrt{2 \mathrm{y}}-3=0\)
3 \(3 x-\sqrt{2 y}+6=0\)
4 \(x-\sqrt{3 y}+2=0\)
Explanation:
B Hyperbola \(\mathrm{x}^2-\mathrm{y}^2=9\)
..(i)
And given chord of contact \(x=9\)
From equation (i) and equation (ii)
Put, \(\quad x=9\) in (i)
\(81-y^2=9\)
\(y^2=72\)
\(y= \pm 6 \sqrt{2}\)
Now, equation of tangent at \(\mathrm{P}=\mathrm{P}=(9,6 \sqrt{2})\) is \(\mathrm{S}_1=0\)
\(\Rightarrow \quad \mathrm{x}(9)-\mathrm{y}(6 \sqrt{2})-9=0\)
\(3 \mathrm{x}-2 \sqrt{2 \mathrm{y}}-3=0\)
AP EAMCET-2013
Hyperbola
120795
The foci of the ellipse \(\frac{\mathrm{x}^2}{16}+\frac{y^2}{b^2}=1\) and the hyperbola \(\frac{x^2}{144}-\frac{y^2}{81}=\frac{1}{25}\) coincide. Then, the value of \(b^2\) is
120793
If the circle \(x^2+y^2=a^2\) intersects the hyperbola \(x y=c^2\) in four points \(\left(x_i, y_i\right)\) for \(i=1,2,3\) and 4 , then \(\mathbf{y}_1+\mathbf{y}_2+\mathbf{y}_3+\mathbf{y}_4\) equals
1 0
2 \(\mathrm{c}\)
3 a
4 \(\mathrm{c}^4\)
Explanation:
A Given, circle : \(x^2+y^2=a^2\)
\(H: x y=c^2\)
\(\therefore \quad \mathrm{x}^2+\frac{\mathrm{c}^4}{\mathrm{x}^2}=\mathrm{a}^2\)
\(x^4-a^2 x^2+c^4=0\)
\(\mathrm{x}_1+\mathrm{x}_2+\mathrm{x}_3+\mathrm{x}_4=0\)
\(\text { or } \mathrm{y}_1+\mathrm{y}_2+\mathrm{y}_3+\mathrm{y}_4=0\)
Now, sum of roots:
AP EAMCET-2009
Hyperbola
120794
If \(x=9\) is a chord of contact of the hyperbola \(x^2\) \(-y^2=9\), then the equation of the tangent at one of the points of contact is
1 \(x+\sqrt{3 y}+2=0\)
2 \(3 \mathrm{x}-2 \sqrt{2 \mathrm{y}}-3=0\)
3 \(3 x-\sqrt{2 y}+6=0\)
4 \(x-\sqrt{3 y}+2=0\)
Explanation:
B Hyperbola \(\mathrm{x}^2-\mathrm{y}^2=9\)
..(i)
And given chord of contact \(x=9\)
From equation (i) and equation (ii)
Put, \(\quad x=9\) in (i)
\(81-y^2=9\)
\(y^2=72\)
\(y= \pm 6 \sqrt{2}\)
Now, equation of tangent at \(\mathrm{P}=\mathrm{P}=(9,6 \sqrt{2})\) is \(\mathrm{S}_1=0\)
\(\Rightarrow \quad \mathrm{x}(9)-\mathrm{y}(6 \sqrt{2})-9=0\)
\(3 \mathrm{x}-2 \sqrt{2 \mathrm{y}}-3=0\)
AP EAMCET-2013
Hyperbola
120795
The foci of the ellipse \(\frac{\mathrm{x}^2}{16}+\frac{y^2}{b^2}=1\) and the hyperbola \(\frac{x^2}{144}-\frac{y^2}{81}=\frac{1}{25}\) coincide. Then, the value of \(b^2\) is
120793
If the circle \(x^2+y^2=a^2\) intersects the hyperbola \(x y=c^2\) in four points \(\left(x_i, y_i\right)\) for \(i=1,2,3\) and 4 , then \(\mathbf{y}_1+\mathbf{y}_2+\mathbf{y}_3+\mathbf{y}_4\) equals
1 0
2 \(\mathrm{c}\)
3 a
4 \(\mathrm{c}^4\)
Explanation:
A Given, circle : \(x^2+y^2=a^2\)
\(H: x y=c^2\)
\(\therefore \quad \mathrm{x}^2+\frac{\mathrm{c}^4}{\mathrm{x}^2}=\mathrm{a}^2\)
\(x^4-a^2 x^2+c^4=0\)
\(\mathrm{x}_1+\mathrm{x}_2+\mathrm{x}_3+\mathrm{x}_4=0\)
\(\text { or } \mathrm{y}_1+\mathrm{y}_2+\mathrm{y}_3+\mathrm{y}_4=0\)
Now, sum of roots:
AP EAMCET-2009
Hyperbola
120794
If \(x=9\) is a chord of contact of the hyperbola \(x^2\) \(-y^2=9\), then the equation of the tangent at one of the points of contact is
1 \(x+\sqrt{3 y}+2=0\)
2 \(3 \mathrm{x}-2 \sqrt{2 \mathrm{y}}-3=0\)
3 \(3 x-\sqrt{2 y}+6=0\)
4 \(x-\sqrt{3 y}+2=0\)
Explanation:
B Hyperbola \(\mathrm{x}^2-\mathrm{y}^2=9\)
..(i)
And given chord of contact \(x=9\)
From equation (i) and equation (ii)
Put, \(\quad x=9\) in (i)
\(81-y^2=9\)
\(y^2=72\)
\(y= \pm 6 \sqrt{2}\)
Now, equation of tangent at \(\mathrm{P}=\mathrm{P}=(9,6 \sqrt{2})\) is \(\mathrm{S}_1=0\)
\(\Rightarrow \quad \mathrm{x}(9)-\mathrm{y}(6 \sqrt{2})-9=0\)
\(3 \mathrm{x}-2 \sqrt{2 \mathrm{y}}-3=0\)
AP EAMCET-2013
Hyperbola
120795
The foci of the ellipse \(\frac{\mathrm{x}^2}{16}+\frac{y^2}{b^2}=1\) and the hyperbola \(\frac{x^2}{144}-\frac{y^2}{81}=\frac{1}{25}\) coincide. Then, the value of \(b^2\) is
120793
If the circle \(x^2+y^2=a^2\) intersects the hyperbola \(x y=c^2\) in four points \(\left(x_i, y_i\right)\) for \(i=1,2,3\) and 4 , then \(\mathbf{y}_1+\mathbf{y}_2+\mathbf{y}_3+\mathbf{y}_4\) equals
1 0
2 \(\mathrm{c}\)
3 a
4 \(\mathrm{c}^4\)
Explanation:
A Given, circle : \(x^2+y^2=a^2\)
\(H: x y=c^2\)
\(\therefore \quad \mathrm{x}^2+\frac{\mathrm{c}^4}{\mathrm{x}^2}=\mathrm{a}^2\)
\(x^4-a^2 x^2+c^4=0\)
\(\mathrm{x}_1+\mathrm{x}_2+\mathrm{x}_3+\mathrm{x}_4=0\)
\(\text { or } \mathrm{y}_1+\mathrm{y}_2+\mathrm{y}_3+\mathrm{y}_4=0\)
Now, sum of roots:
AP EAMCET-2009
Hyperbola
120794
If \(x=9\) is a chord of contact of the hyperbola \(x^2\) \(-y^2=9\), then the equation of the tangent at one of the points of contact is
1 \(x+\sqrt{3 y}+2=0\)
2 \(3 \mathrm{x}-2 \sqrt{2 \mathrm{y}}-3=0\)
3 \(3 x-\sqrt{2 y}+6=0\)
4 \(x-\sqrt{3 y}+2=0\)
Explanation:
B Hyperbola \(\mathrm{x}^2-\mathrm{y}^2=9\)
..(i)
And given chord of contact \(x=9\)
From equation (i) and equation (ii)
Put, \(\quad x=9\) in (i)
\(81-y^2=9\)
\(y^2=72\)
\(y= \pm 6 \sqrt{2}\)
Now, equation of tangent at \(\mathrm{P}=\mathrm{P}=(9,6 \sqrt{2})\) is \(\mathrm{S}_1=0\)
\(\Rightarrow \quad \mathrm{x}(9)-\mathrm{y}(6 \sqrt{2})-9=0\)
\(3 \mathrm{x}-2 \sqrt{2 \mathrm{y}}-3=0\)
AP EAMCET-2013
Hyperbola
120795
The foci of the ellipse \(\frac{\mathrm{x}^2}{16}+\frac{y^2}{b^2}=1\) and the hyperbola \(\frac{x^2}{144}-\frac{y^2}{81}=\frac{1}{25}\) coincide. Then, the value of \(b^2\) is
