Question

A normal is drawn to parabola y^2 =4x at (1,2) and tangent drawn to y= e^x at (c, e) if tangent and normal intersect at x- axis then find c​

Answer 1

A normal is drawn to parabola y^2 = 4x at (1,2) and tangent drawn to y= e^x at (c, e^c).

To find : if tangent and normal intersect at x - axis we have to find the value of c.

solution : slope of tangent of parabola = dy/dx = 2/y

at (1, 2), slope of tangent , m = 2/2 = 1

now slope of normal = -1/slope of tangent = -1

equation of normal is ...

(y - 2) = -1(x - 1) ⇒x + y = 3 it intersects at the point (3, 0) on x - axis.

so, equation of tangent of y = e^x at (3, 0)

slope of tangent = dy/dx = e^x

at (c, e^c) , slope of tangent = e^c

now equation of tangent passing through (c, e^c) is ...

(y - e^c) = e^c(x - c)

putting (3, 0)

so, 0 - e^c = e^c(3 - c)

⇒-1 = 3 - c

⇒c = 4

Therefore the value of c = 4