K. C. Sinha Solutions for Chapter: Circle, Exercise 1: 6.1

If $11a^2+2b^2+2ak+1=0$, where $k$ is a fixed real number, then the value of $k^4$ for which $ax+by+1=0$ is tangent to a fixed circle is ...

The extremities of a diagonal of a rectangle are $\left(-4,4\right)$ and $\left(6,-1\right)$. A circle circumscribes the rectangle and cuts an intercept $AB$ on the $y$-axis. If $\Delta$ be the area of the triangle formed by $AB$ and the tangents to the circle at $A$ and $B$, then $8\Delta =$.

A rectangle $ABCD$ is inscribed in a circle with a diameter lying along the line $3y=x+10$. If $A$ and $B$ are the points $\left(-6,7\right)$ and $\left(4,7\right)$ respectively, then area of rectangle $ABCD$ in square units is ....

If curves $a_1{x}^2+2h_{1}xy+b_1{y}^2-2g_{1}x-2f_{1}y+c=0$ and $a_2{x}^2-2h_{2}xy+\left(a_2+a_1-b_1\right)y^2-2g_{2}x-2f_{2}y+c=0$ intersect in four concyclic points$A,B,C$ and $D$ and $H$ be the point $\left(\frac{g_1+g_2}{a_1+a_2},\frac{f_1+f_2}{a_1+a_2}\right)$, then $\frac{5H{A}^2+7H{B}^2+8H{C}^2}{H{D}^2}=\dots .$

The least positive integral value of $\lambda$ for which two chords bisected by $x$-axis can be drawn to the circle $x^2+y^2-\lambda ax-ay-a^2\left({\lambda }^2+1\right)=0$ from point $\left(a\left(\lambda -1\right),a\left(\lambda +1\right)\right)$ is ...

Number of integral values of $\lambda$ for which the variable line $3x+4y-\lambda =0$ lies between the circles $x^2+y^2-2x-2y+1=0$ and $x^2+y^2-18x-2y+78=0$ without intersecting any circle at two distinct points.

If the coordinates of points $A\mathit{,}B\mathit{,}C$ satisfy the relation $xy=1000$ and $\left(\alpha ,\beta \right)$ be the coordinates of the orthocentre of $∆ABC$, then the value of $\alpha \beta$ is

A line of slope $m$ is such that its segment between the lines $5x-y+4=0$ and $3x+4y-4=0$ is bisected at $\left(1,5\right)$, then $3 m=$