Question

Q-The line 4y=X+c, where c is constant is a tangent to curve y^2=X+3 at the point curve. find a value of c​.
(ii) find coordinates of P

Answer 1

The value of c is 7 and the coordinates of point P are (1,2)

Given:

The line $4y = x + c$

The line is tangent to the curve $y^{2} = x + 3$

To find:

The value of c and the coordinates of point P

Solution:

(h,k) be the coordinates of the point P.

The equation of the line is $4y = x + c$

$\frac{1}{4}$  is the slope of the line $4y = x + c$

Also, $y^{2} = x + 3$ is the equation of the curve.

Differentiating both sides of the curve $y^{2} = x + 3$  with respect to x we get

$2y(\frac{dx}{dy} ) = 1$

⇒ $(\frac{dx}{dy} ) = \frac{1}{2y}$

The slope of the line at the point (h,k) is $\frac{1}{2k}$

By the given condition

$\frac{1}{2k} = \frac{1}{4}$

⇒ k = 2

Since (h,k) is the point on the line $y^{2} = x + 3$

⇒ $k^{2} = h + 3$

⇒  $2^{2} = h + 3$

⇒ $h = 1$

The coordinates of the point P are (1,2)

Also, the point (h,k) is on the line $4y = x + c$  

   ⇒ $4k = h + c$

   ⇒ $8 = 1 + c$

   ⇒ $c = 7$

The value of c is 7

The coordinates of point P are (1,2)

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