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Standard XII

Mathematics

Length of Tangent

Normals are d...

Question

Normals are drawn from a point P(h, k) with slopes $m_{1}, m_{2}, m_{3}$ to the parabola $C_{1} : y^{2} = 4 x$

If the curve C is the locus of point P with $m_{1} m_{2} = 2$ , then number of common tangents to the curve $C_{1}$ and curve C is

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infinite

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Solution

The correct option is D
infinite

$m_{1} . m_{2} m_{3} = - k \Rightarrow m_{3} = - \frac{k}{2}$
$\Rightarrow \frac{k^{3}}{8} + (2 - h) (- \frac{k}{3}) \Rightarrow k = 0$
$\Rightarrow y^{2} = 4 x$
$\Rightarrow$ C and $C_{1}$ are same curve
$\Rightarrow$ infinite common tangents

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Similar questions

Q. Normals are drawn from the point

P

with slopes

m_{1}, m_{2}, m_{3}

to the parabola

y^{2} = 4 x .

If the locus of

P

when

m_{1}, m_{2} = α

is part of the parabola itself then the value of

α

is.

Normals are drawn from a point P(h, k) with slopes $m_{1}, m_{2}, m_{3}$ to the parabola $C_{1} : y^{2} = 4 x$

$\sum_{α = 0}^{\infty} (m_{1})^{α} + \sum_{α = 0}^{\infty} (m_{2})^{α} + \sum_{α = 0}^{\infty} (m_{3})^{α} = \frac{89}{22}, w h e r e | m_{i} | < 1 \forall i = 1, 2, 3 t h e n 67 h - 89 k =$

Q. Three normals with slopes

m_{1}, m_{2}

and

m_{3}

are drawn from a point

P

not on the axis of the parabola

y^{2} = 4 x .

m_{1} m_{2} = α

, results in the locus of

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Q. Consider two concentric circles

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

and

C_{2} : x^{2} + y^{2} - 9 = 0.

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axis and having an arbitrary tangent of

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is the curve of locus of focus of drawn parabola and

e

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Q. Let two tangents are drawn to the curve

y^{2} - 4 (x + y) = 3 sin θ + 4 cos θ - 15, x, y, θ \in R

from the origin whose slopes are

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Normals are drawn from a point P(h, k) with slopes m1,m2,m3 to the parabola C1:y2=4x If the curve C is the locus of point P with m1m2=2, then number of common tangents to the curve C1 and curve C is

The correct option is D infinite m1.m2m3=−k⇒m3=−k2 ⇒k38+(2−h)(−k3)⇒k=0 ⇒y2=4x ⇒ C and C1 are same curve ⇒ infinite common tangents

Normals are drawn from a point P(h, k) with slopes $m_{1}, m_{2}, m_{3}$ to the parabola $C_{1} : y^{2} = 4 x$

If the curve C is the locus of point P with $m_{1} m_{2} = 2$ , then number of common tangents to the curve $C_{1}$ and curve C is

The correct option is D
infinite

$m_{1} . m_{2} m_{3} = - k \Rightarrow m_{3} = - \frac{k}{2}$
$\Rightarrow \frac{k^{3}}{8} + (2 - h) (- \frac{k}{3}) \Rightarrow k = 0$
$\Rightarrow y^{2} = 4 x$
$\Rightarrow$ C and $C_{1}$ are same curve
$\Rightarrow$ infinite common tangents