Consider the two curves C -1- y -2-4 x - -C -2- x -2- y -2-6 x -1-0 -Then-A- C 1 and C 2 touch each other only at one pointB- C 1 and C 2 touch each other exactly at two pointsC- C 1 and C 2 neither intersect nor touch each otherD- C 1 and C 2 intersect but do not touch at exactly two points

The correct option is B C_1 and C_2 touch each other exactly at two points nbsp;For points of intersection of circle and parabola, put y^2 = 4x in the equati ...

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Standard XII

Mathematics

Standard Equation of Parabola

Consider the ...

Question

Consider the two curves
$C_{1} : y^{2} = 4 x, C_{2} : x^{2} + y^{2} - 6 x + 1 = 0.$
Then,

C_{1}

and

C_{2}

touch each other only at one point

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C_{1}

and

C_{2}

touch each other exactly at two points

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C_{1}

and

C_{2}

intersect (but do not touch) at exactly two points

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C_{1}

and

C_{2}

neither intersect nor touch each other

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Solution

The correct option is B $C_{1}$ and $C_{2}$ touch each other exactly at two points
For points of intersection of circle and parabola, put $y^{2} = 4 x$ in the equation of the circle, we get

$x^{2} + 4 x - 6 x + 1 = 0 o r (x - 1)^{2} = 0$
i.e., they intersect at x = 1.
After substituting in the equation of parabola, we get
$y = 2, - 2$
Thus they touch at the above two points.

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Similar questions

Q. Consider the two curves

C_{1} : y^{2} = 4 x, C_{2} : x^{2} + y^{2} - 6 x + 1 = 0.

Then,

Q. The curve

C_{1} : y = e^{- x}

and

C_{2} : y = e^{- x} sin x (x > 0)

touches each other at infinitely many points. Let

0 < x_{1} < x_{2} < \dots < x_{n} < \dots

be the abscissa of these points of contact. If

A_{n}

denotes the area bounded by two curves & ordinates

x = x_{n}

and

x = x_{n + 1}

, then

Q. Let

C_{1}, C_{2}

be two circles touching each other externally at the point A and let AB be the diameter of circle

C_{1}

. Draw a secant

B A_{3}

to circle

C_{2}

, intersecting circle

C_{1}

at a point

A_{1} (\neq A)

, and circle

C_{2}

at points

A_{2}

and

A_{3}

. If

B A_{1} = 2

B A_{2} = 3

and

B A_{3} = 4

, then the radii of circles

C_{1}

and

C_{2}

are respectively.

Q. Assertion :The circles

x^{2} + y^{2} + 2 a x + c = 0

x^{2} + y^{2} + 2 b y + c = 0

touch each other if

\frac{1}{c} = \frac{1}{a^{2}} + \frac{1}{b^{2}}

. Reason: Two cirlces with centre

c_{1}, c_{2}

& radii

r_{1}, r_{2}

touch each other if

c_{1} + c_{2} = r_{1} \pm r_{2}

Q. Circles

C_{1}

and

C_{2}

, of radii r and R respectively, touch each other as shown in figure. The line l, which is parallel to the line joining the centres of

C_{1}

and

C_{2}

, is tangent to

C_{1}

at P and intersects

C_{2}

at A,B. If

R^{2} = 2 r^{2}

, then

∠ A O B

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Consider the two curves C1:y2=4x,C2:x2+y2−6x+1=0. Then,

Consider the two curves
$C_{1} : y^{2} = 4 x, C_{2} : x^{2} + y^{2} - 6 x + 1 = 0.$
Then,