The centre of the circle passing through the | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) The tangent to the parabola $y = x^2$ at $(2, 4)$ is given by $\frac{1}{2}(y + 4) = x(2)$,which simplifies to $4x - y - 4 = 0$.Since the circle to

Vedclass · Accepted Answer

$\left( \frac{-16}{5}, \frac{53}{10} \right)$

Vedclass · Answer

(C) The tangent to the parabola $y = x^2$ at $(2, 4)$ is given by $\frac{1}{2}(y + 4) = x(2)$,which simplifies to $4x - y - 4 = 0$.Since the circle touches the parabola at $(2, 4)$,the centre $(h, k)$ of the circle must lie on the normal to the parabola at $(2, 4)$.The slope of the tangent is $4$,so the slope of the normal is $-\frac{1}{4}$.The equation of the normal at $(2, 4)$ is $y - 4 = -\frac{1}{4}(x - 2)$,which simplifies to $x + 4y = 18$.Thus,$h + 4k = 18$ $(i)$.Since the circle passes through $(0, 1)$ and $(2, 4)$,the distance from the centre $(h, k)$ to these points must be equal:$(h - 2)^2 + (k - 4)^2 = (h - 0)^2 + (k - 1)^2$.Expanding this,$h^2 - 4h + 4 + k^2 - 8k + 16 = h^2 + k^2 - 2k + 1$,which simplifies to $4h + 6k = 19$ $(ii)$.Solving equations $(i)$ and $(ii)$ simultaneously: from $(i)$,$h = 18 - 4k$. Substituting into $(ii)$ gives $4(18 - 4k) + 6k = 19$,so $72 - 16k + 6k = 19$,which means $10k = 53$,so $k = \frac{53}{10}$.Then $h = 18 - 4(\frac{53}{10}) = 18 - \frac{106}{5} = \frac{90 - 106}{5} = -\frac{16}{5}$.The centre is $\left( -\frac{16}{5}, \frac{53}{10} \right)$.

The centre of the circle passing through the point $(0, 1)$ and touching the curve $y = x^2$ at $(2, 4)$ is

Similar Questions

If $(x, 3)$ and $(3, 5)$ are the extremities of a diameter of a circle with centre at $(2, y)$,then the values of $x$ and $y$ are:

The length of the chord joining the points $(4 \cos \theta, 4 \sin \theta)$ and $(4 \cos (\theta+60^{\circ}), 4 \sin (\theta+60^{\circ}))$ of the circle $x^{2}+y^{2}=16$ is

If one end of the diameter of the circle $x^2+y^2-4x-6y+11=0$ is $(3,4)$,then the other end of the diameter is

Find the equation of the pair of straight lines parallel to the $x$-axis and touching the circle $x^2 + y^2 - 6x - 4y - 12 = 0$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module