120240 If the tangents and normals at the extremities of a focal chord of a parabola intersect at \(\left(x_1, y_2\right)\) and \(\left(x_2, y_2\right)\) respectively, then

120241 Let \(P(a, b)\) be a point on the parabola \(y^2=8 x\) such that the tangent at \(P\) passes through the centre of the circle \(x^2+y^2-10 x-14 y+65=0\). Let \(A\) be the product of all possible values of a and \(b\) be the product of all possible values of \(b\). Then the value of \(A+B\) is equal to

120242 Let the tangent drawn to the parabola \(y^2=24 x\) at the point \((\alpha, \beta)\) is perpendicular to the line \(2 x+2 y=5\). Then the normal to the hyperbola \(\frac{x^2}{\alpha^2}-\frac{y^2}{\beta^2}=1\) at the point \((\alpha+4, \beta+4)\) does NOT pass through the point :

120243 Consider ellipses \(\mathrm{E}_{\mathrm{k}}: \mathrm{kx}^2+\mathrm{k}^2 \mathbf{y}^2=\mathbf{1}, \mathrm{k}=\mathbf{1}, \mathbf{2}\), \(\ldots ., 20\). Let \(\mathrm{C}_{\mathrm{k}}\) be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse \(E_k\), If \(r_k\) is the radius of the circle \(C_k\), then the value of \(\sum_{\mathrm{k}=1}^{20} \frac{1}{\mathrm{r}_{\mathrm{k}}^2}\) is

120240 If the tangents and normals at the extremities of a focal chord of a parabola intersect at \(\left(x_1, y_2\right)\) and \(\left(x_2, y_2\right)\) respectively, then

120241 Let \(P(a, b)\) be a point on the parabola \(y^2=8 x\) such that the tangent at \(P\) passes through the centre of the circle \(x^2+y^2-10 x-14 y+65=0\). Let \(A\) be the product of all possible values of a and \(b\) be the product of all possible values of \(b\). Then the value of \(A+B\) is equal to

120242 Let the tangent drawn to the parabola \(y^2=24 x\) at the point \((\alpha, \beta)\) is perpendicular to the line \(2 x+2 y=5\). Then the normal to the hyperbola \(\frac{x^2}{\alpha^2}-\frac{y^2}{\beta^2}=1\) at the point \((\alpha+4, \beta+4)\) does NOT pass through the point :

120243 Consider ellipses \(\mathrm{E}_{\mathrm{k}}: \mathrm{kx}^2+\mathrm{k}^2 \mathbf{y}^2=\mathbf{1}, \mathrm{k}=\mathbf{1}, \mathbf{2}\), \(\ldots ., 20\). Let \(\mathrm{C}_{\mathrm{k}}\) be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse \(E_k\), If \(r_k\) is the radius of the circle \(C_k\), then the value of \(\sum_{\mathrm{k}=1}^{20} \frac{1}{\mathrm{r}_{\mathrm{k}}^2}\) is

120240 If the tangents and normals at the extremities of a focal chord of a parabola intersect at \(\left(x_1, y_2\right)\) and \(\left(x_2, y_2\right)\) respectively, then

120241 Let \(P(a, b)\) be a point on the parabola \(y^2=8 x\) such that the tangent at \(P\) passes through the centre of the circle \(x^2+y^2-10 x-14 y+65=0\). Let \(A\) be the product of all possible values of a and \(b\) be the product of all possible values of \(b\). Then the value of \(A+B\) is equal to

120242 Let the tangent drawn to the parabola \(y^2=24 x\) at the point \((\alpha, \beta)\) is perpendicular to the line \(2 x+2 y=5\). Then the normal to the hyperbola \(\frac{x^2}{\alpha^2}-\frac{y^2}{\beta^2}=1\) at the point \((\alpha+4, \beta+4)\) does NOT pass through the point :

120243 Consider ellipses \(\mathrm{E}_{\mathrm{k}}: \mathrm{kx}^2+\mathrm{k}^2 \mathbf{y}^2=\mathbf{1}, \mathrm{k}=\mathbf{1}, \mathbf{2}\), \(\ldots ., 20\). Let \(\mathrm{C}_{\mathrm{k}}\) be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse \(E_k\), If \(r_k\) is the radius of the circle \(C_k\), then the value of \(\sum_{\mathrm{k}=1}^{20} \frac{1}{\mathrm{r}_{\mathrm{k}}^2}\) is

120240 If the tangents and normals at the extremities of a focal chord of a parabola intersect at \(\left(x_1, y_2\right)\) and \(\left(x_2, y_2\right)\) respectively, then

120241 Let \(P(a, b)\) be a point on the parabola \(y^2=8 x\) such that the tangent at \(P\) passes through the centre of the circle \(x^2+y^2-10 x-14 y+65=0\). Let \(A\) be the product of all possible values of a and \(b\) be the product of all possible values of \(b\). Then the value of \(A+B\) is equal to

120242 Let the tangent drawn to the parabola \(y^2=24 x\) at the point \((\alpha, \beta)\) is perpendicular to the line \(2 x+2 y=5\). Then the normal to the hyperbola \(\frac{x^2}{\alpha^2}-\frac{y^2}{\beta^2}=1\) at the point \((\alpha+4, \beta+4)\) does NOT pass through the point :

120243 Consider ellipses \(\mathrm{E}_{\mathrm{k}}: \mathrm{kx}^2+\mathrm{k}^2 \mathbf{y}^2=\mathbf{1}, \mathrm{k}=\mathbf{1}, \mathbf{2}\), \(\ldots ., 20\). Let \(\mathrm{C}_{\mathrm{k}}\) be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse \(E_k\), If \(r_k\) is the radius of the circle \(C_k\), then the value of \(\sum_{\mathrm{k}=1}^{20} \frac{1}{\mathrm{r}_{\mathrm{k}}^2}\) is