Question

A curve has equation y = 2x² + 3x - 2. The tangent to the curve at x = 4 meets the x-axis at the point A. Find the coordinates of the point A.​

Answer 1

$\large\underline{\sf{Solution-}}$

Given curve is

$\rm :\longmapsto\:y = {2x}^{2} + 3x - 2$

The tangent touches the curve at a point whose x - coordinate is 4.

So, substituting x = 4 in given curve we get

$\rm :\longmapsto\:y = {2(4)}^{2} + 3(4) - 2$

$\rm :\longmapsto\:y = 32 + 12 - 2$

$\bf\implies \:y = 42$

So, coordinates of point of contact of tangent and curve is P ( 4, 42 ).

Now,

Given curve is

$\rm :\longmapsto\:y = {2x}^{2} + 3x - 2$

On differentiating both sides w. r. t. x, we get

$\rm :\longmapsto\:\dfrac{d}{dx}y = \dfrac{d}{dx}( {2x}^{2} + 3x - 2)$

$\rm :\longmapsto\:\dfrac{dy}{dx}= 2\dfrac{d}{dx}{x}^{2} + 3\dfrac{d}{dx}x - \dfrac{d}{dx}2$

$\rm :\longmapsto\:\dfrac{dy}{dx}= 4x + 3$

So, slope of tangent at point P is given by

$\rm :\longmapsto\:Slope \: of \: Tangent, \: m = \bigg(\dfrac{dy}{dx}\bigg)_P$

$\rm \: = \: \: 4(4) + 3$

$\rm \: = \: \: 16 + 3$

$\rm \: = \: \: 19$

$\bf\implies \:Slope \: of \: Tangent, \: m = 19$

Now,

We know that,

Equation of tangent line passes through the point ( a, b ) having slope m is given by

$\red{ \boxed{ \bf{ \: y \: - \: b \: = \: m \: ( \: x \: - \: a \: )}}}$

So,

Equation of tangent line passes through the point (4, 42) having slope m = 19, is

$\rm :\longmapsto\:y - 42 = 19(x - 4)$

$\rm :\longmapsto\:y - 42 = 19x - 76$

$\rm :\longmapsto\:19x - y - 34 = 0$

Now, this tangent line meet the x - axis at A.

We know, On x - axis, y = 0

So, on substituting y = 0, we get

$\rm :\longmapsto\:19x - 0 - 34 = 0$

$\rm :\longmapsto\:19x = 34$

$\bf\implies \:x = \dfrac{34}{19}$

So,

$\rm :\longmapsto\:Coordinates \: of \: A = \bigg(\dfrac{34}{19}, \: 0\bigg)$

Additional Information

1. Let y = f(x) be any curve, then line which touches the curve y = f(x) exactly at one point say P is called tangent and that very point P, if we draw a perpendicular on tangent, that line is called normal to the curve at P.

2. If tangent is parallel to x - axis, its slope is 0.

3. If tangent is parallel to y - axis, its slope is not defined.

4. Two lines having slope M and m are parallel, iff M = m.

5. If two lines having slope M and m are perpendicular, iff Mm = - 1.