The intercept on the line $y = x$ by the circle ${x^2} + {y^2} - 2x = 0$ is $AB$ . Equation of the circle with $AB$ as a diameter is
The intercept on the line $y = x$ by the circle ${x^2} + {y^2} - 2x = 0$ is $AB$ . Equation of the circle with $AB$ as a diameter is
- [IIT 1996]
- A
${x^2} + {y^2} - x - y = 0$
- B
${x^2} + {y^2} - 2x - y = 0$
- C
${x^2} + {y^2} - x + y = 0$
- D
${x^2} + {y^2} + x - y = 0$
Similar Questions
Let $C: x^2+y^2=4$ and $C^{\prime}: x^2+y^2-4 \lambda x+9=0$ be two circles. If the set of all values of $\lambda$ so that the circles $\mathrm{C}$ and $\mathrm{C}^{\prime}$ intersect at two distinct points, is ${R}-[a, b]$, then the point $(8 a+12,16 b-20)$ lies on the curve:
Let $C: x^2+y^2=4$ and $C^{\prime}: x^2+y^2-4 \lambda x+9=0$ be two circles. If the set of all values of $\lambda$ so that the circles $\mathrm{C}$ and $\mathrm{C}^{\prime}$ intersect at two distinct points, is ${R}-[a, b]$, then the point $(8 a+12,16 b-20)$ lies on the curve:
- [JEE MAIN 2024]
A circle $S$ passes through the point $(0,1)$ and is orthogonal to the circles $(x-1)^2+y^2=16$ and $x^2+y^2=1$. Then
$(A)$ radius of $S$ is $8$
$(B)$ radius of $S$ is $7$
$(C)$ centre of $S$ is $(-7,1)$
$(D)$ centre of $S$ is $(-8,1)$
A circle $S$ passes through the point $(0,1)$ and is orthogonal to the circles $(x-1)^2+y^2=16$ and $x^2+y^2=1$. Then
$(A)$ radius of $S$ is $8$
$(B)$ radius of $S$ is $7$
$(C)$ centre of $S$ is $(-7,1)$
$(D)$ centre of $S$ is $(-8,1)$
- [IIT 2014]
Answer the following by appropriately matching the lists based on the information given in the paragraph
Let the circles $C_1: x^2+y^2=9$ and $C_2:(x-3)^2+(y-4)^2=16$, intersect at the points $X$ and $Y$. Suppose that another circle $C_3:(x-h)^2+(y-k)^2=r^2$ satisfies the following conditions :
$(i)$ centre of $C _3$ is collinear with the centres of $C _1$ and $C _2$
$(ii)$ $C _1$ and $C _2$ both lie inside $C _3$, and
$(iii)$ $C _3$ touches $C _1$ at $M$ and $C _2$ at $N$.
Let the line through $X$ and $Y$ intersect $C _3$ at $Z$ and $W$, and let a common tangent of $C _1$ and $C _3$ be a tangent to the parabola $x^2=8 \alpha y$.
There are some expression given in the $List-I$ whose values are given in $List-II$ below:
$List-I$
$List-II$
$(I)$ $2 h + k$
$(P)$ $6$
$(II)$ $\frac{\text { Length of } ZW }{\text { Length of } XY }$
$(Q)$ $\sqrt{6}$
$(III)$ $\frac{\text { Area of triangle } MZN }{\text { Area of triangle ZMW }}$
$(R)$ $\frac{5}{4}$
$(IV)$ $\alpha$
$(S)$ $\frac{21}{5}$
$(T)$ $2 \sqrt{6}$
$(U)$ $\frac{10}{3}$
($1$) Which of the following is the only INCORRECT combination?
$(1) (IV), (S)$ $(2) (IV), (U)$ $(3) (III), (R)$ $(4) (I), (P)$
($2$) Which of the following is the only CORRECT combination?
$(1) (II), (T)$ $(2) (I), (S)$ $(3) (I), (U)$ $(4) (II), (Q)$
Give the answer or quetion ($1$) and ($2$)
Answer the following by appropriately matching the lists based on the information given in the paragraph
Let the circles $C_1: x^2+y^2=9$ and $C_2:(x-3)^2+(y-4)^2=16$, intersect at the points $X$ and $Y$. Suppose that another circle $C_3:(x-h)^2+(y-k)^2=r^2$ satisfies the following conditions :
$(i)$ centre of $C _3$ is collinear with the centres of $C _1$ and $C _2$
$(ii)$ $C _1$ and $C _2$ both lie inside $C _3$, and
$(iii)$ $C _3$ touches $C _1$ at $M$ and $C _2$ at $N$.
Let the line through $X$ and $Y$ intersect $C _3$ at $Z$ and $W$, and let a common tangent of $C _1$ and $C _3$ be a tangent to the parabola $x^2=8 \alpha y$.
There are some expression given in the $List-I$ whose values are given in $List-II$ below:
| $List-I$ | $List-II$ |
| $(I)$ $2 h + k$ | $(P)$ $6$ |
| $(II)$ $\frac{\text { Length of } ZW }{\text { Length of } XY }$ | $(Q)$ $\sqrt{6}$ |
| $(III)$ $\frac{\text { Area of triangle } MZN }{\text { Area of triangle ZMW }}$ | $(R)$ $\frac{5}{4}$ |
| $(IV)$ $\alpha$ | $(S)$ $\frac{21}{5}$ |
| $(T)$ $2 \sqrt{6}$ | |
| $(U)$ $\frac{10}{3}$ |
($1$) Which of the following is the only INCORRECT combination?
$(1) (IV), (S)$ $(2) (IV), (U)$ $(3) (III), (R)$ $(4) (I), (P)$
($2$) Which of the following is the only CORRECT combination?
$(1) (II), (T)$ $(2) (I), (S)$ $(3) (I), (U)$ $(4) (II), (Q)$
Give the answer or quetion ($1$) and ($2$)
- [IIT 2019]
The minimum distance between any two points $P _{1}$ and $P _{2}$ while considering point $P _{1}$ on one circle and point $P _{2}$ on the other circle for the given circles' equations
$x^{2}+y^{2}-10 x-10 y+41=0$
$x^{2}+y^{2}-24 x-10 y+160=0$ is .........
The minimum distance between any two points $P _{1}$ and $P _{2}$ while considering point $P _{1}$ on one circle and point $P _{2}$ on the other circle for the given circles' equations
$x^{2}+y^{2}-10 x-10 y+41=0$
$x^{2}+y^{2}-24 x-10 y+160=0$ is .........
- [JEE MAIN 2021]
If one of the diameters of the circle $x^{2}+y^{2}-2 \sqrt{2} x$ $-6 \sqrt{2} y+14=0$ is a chord of the circle $(x-2 \sqrt{2})^{2}$ $+(y-2 \sqrt{2})^{2}=r^{2}$, then the value of $r^{2}$ is equal to
If one of the diameters of the circle $x^{2}+y^{2}-2 \sqrt{2} x$ $-6 \sqrt{2} y+14=0$ is a chord of the circle $(x-2 \sqrt{2})^{2}$ $+(y-2 \sqrt{2})^{2}=r^{2}$, then the value of $r^{2}$ is equal to
- [JEE MAIN 2022]