Question

The points of contact q and r of tangent from the point p(2,3) on the parabola y^2=4x are

Answer 1

Answer:

Step-by-step explanation:

Equation of the parabola

$y^2=4x$

comparing this with $y^2=4ax$

4a=4

a=1

The equation of any tangent is of the form

$y=mx+\frac{a}{m}$

since it passes through(2,3)

$3=2m+\frac{1}{m}$

$3=\frac{2m^2+1}{m}$

$3m=2m^2+1$

$2m^2-3m+1=0$

(m-1)(2m-1)=0

m=1, 1/2

$The\:point\:of\:contact\:is\:$

$(\frac{a}{m^{2}},\frac{2a}{m})$

when m=1,

$The\:point\:of\:contact\:is\:$

$(\frac{1}{1},\frac{2(1)}{1})$

(1,2)

when m=1/2,

$The\:point\:of\:contact\:is\:$

$(\frac{1}{1/4},\frac{2(1)}{1/2})$

(4,4)

Answer 2

Answer:

Slope of tangent line (y  

2

=4x):  2y  

dx

dy

​  

=4

dx

dy

​  

=2  

/y

​  

 

Equation of tangent P(2,3) ⇒y−y  

1

​  

=M(x−x  

1

​  

)

y−3=  

y

2

​  

(x−2)

⇒y  

2

−3y=2x−4 { we know y  

2

=y,x=y  

/4

2

​  

 }

⇒y  

2

−3y=2(  

4

y  

2

 

​  

)−4

⇒y  

2

−  

2

y  

2

 

​  

−3y+4=0

⇒y  

2

−6y+8=0

⇒y  

2

−4y−2y+8=0

⇒y(y−4)−2(y−4)

⇒y  

1

​  

=2,4 {(x=y  

/4

2

​  

)}

∴ Points are (1,2) & (4,4).