Advertisements
Advertisements
Question
The equations of tangent at those points where the curve y = x2 − 3x + 2 meets x-axis are _______________ .
Options
x − y + 2 = 0 = x − y − 1
x + y − 1 = 0 = x − y − 2
x − y − 1 = 0 = x − y
x − y = 0 = x + y
Advertisements
Solution
`x + y − 1 = 0 = x − y − 2`
Let the tangent meet the x-axis at point (x, 0).
Now,
\[y = x^2 - 3x + 2\]
\[ \Rightarrow \frac{dy}{dx} = 2x - 3\]
\[\text { The tangent passes through point (x, 0) }.\]
\[ \therefore 0 = x^2 - 3x + 2\]
\[ \Rightarrow \left( x - 2 \right)\left( x - 1 \right) = 0\]
\[ \Rightarrow x = 2 \ or \ x = 1\]
\[\text { Case 1: When } x=2:\]
\[\text { Slope of the tangent },m= \left( \frac{dy}{dx} \right)_\left( 2, 0 \right) =4-3=1\]
\[ \therefore \left( x_1 , y_1 \right) = \left( 2, 0 \right)\]
\[\text { Equation of the tangent }:\]
\[y - y_1 = m \left( x - x_1 \right)\]
\[ \Rightarrow y - 0 = 1 \left( x - 2 \right)\]
\[ \Rightarrow x - y - 2 = 0\]
\[\text { Case 2: When } x=1:\]
\[\text { Slope of the tangent },m= \left( \frac{dy}{dx} \right)_\left( 2, 0 \right) =2-3=-1\]
\[ \therefore \left( x_1 , y_1 \right) = \left( 1, 0 \right)\]
\[\text { Equation of the tangent }:\]
\[y - y_1 = m \left( x - x_1 \right)\]
\[ \Rightarrow y - 0 = - 1 \left( x - 1 \right)\]
\[ \Rightarrow x + y - 1 = 0\]
APPEARS IN
RELATED QUESTIONS
Find the equations of the tangent and normal to the curve x = a sin3θ and y = a cos3θ at θ=π/4.
Show that the equation of normal at any point t on the curve x = 3 cos t – cos3t and y = 3 sin t – sin3t is 4 (y cos3t – sin3t) = 3 sin 4t
Find the point on the curve y = x3 − 11x + 5 at which the tangent is y = x − 11.
Find the equation of all lines having slope 2 which are tangents to the curve `y = 1/(x- 3), x != 3`
Find the equation of the tangent line to the curve y = x2 − 2x + 7 which is perpendicular to the line 5y − 15x = 13.
Find the points on the curve y = x3 at which the slope of the tangent is equal to the y-coordinate of the point.
Find the equation of the normal at the point (am2, am3) for the curve ay2 = x3.
Find the equation of the tangent to the curve `y = sqrt(3x-2)` which is parallel to the line 4x − 2y + 5 = 0.
Find the equation of the normal to curve y2 = 4x at the point (1, 2).
Find the slope of the tangent and the normal to the following curve at the indicted point \[y = \sqrt{x} \text { at }x = 9\] ?
Find the slope of the tangent and the normal to the following curve at the indicted point x2 + 3y + y2 = 5 at (1, 1) ?
At what point of the curve y = x2 does the tangent make an angle of 45° with the x-axis?
At what points on the curve y = 2x2 − x + 1 is the tangent parallel to the line y = 3x + 4?
At what points on the curve y = x2 − 4x + 5 is the tangent perpendicular to the line 2y + x = 7?
Who that the tangents to the curve y = 7x3 + 11 at the points x = 2 and x = −2 are parallel ?
Find the points on the curve y = x3 where the slope of the tangent is equal to the x-coordinate of the point ?
Find the equation of the normal to y = 2x3 − x2 + 3 at (1, 4) ?
Find the equation of the tangent and the normal to the following curve at the indicated point y = x4 − bx3 + 13x2 − 10x + 5 at (0, 5) ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[y^2 = \frac{x^3}{4 - x}at \left( 2, - 2 \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point y = x2 + 4x + 1 at x = 3 ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( a\sec\theta, b\tan\theta \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated points \[x = \frac{2 a t^2}{1 + t^2}, y = \frac{2 a t^3}{1 + t^2}\text { at } t = \frac{1}{2}\] ?
Find the equation of the tangent and the normal to the following curve at the indicated points x = a(θ + sinθ), y = a(1 − cosθ) at θ ?
Find the equation of the tangent line to the curve y = x2 + 4x − 16 which is parallel to the line 3x − y + 1 = 0 ?
Find an equation of normal line to the curve y = x3 + 2x + 6 which is parallel to the line x+ 14y + 4 = 0 ?
Find the angle of intersection of the following curve x2 + 4y2 = 8 and x2 − 2y2 = 2 ?
Show that the following curve intersect orthogonally at the indicated point x2 = y and x3 + 6y = 7 at (1, 1) ?
Show that the curves \[\frac{x^2}{a^2 + \lambda_1} + \frac{y^2}{b^2 + \lambda_1} = 1 \text { and } \frac{x^2}{a^2 + \lambda_2} + \frac{y^2}{b^2 + \lambda_2} = 1\] intersect at right angles ?
Find the slope of the tangent to the curve x = t2 + 3t − 8, y = 2t2 − 2t − 5 at t = 2 ?
If the curves y = 2 ex and y = ae−x intersect orthogonally, then a = _____________ .
The normal at the point (1, 1) on the curve 2y + x2 = 3 is _____________ .
The equation of the normal to the curve y = sinx at (0, 0) is ______.
Prove that the curves y2 = 4x and x2 + y2 – 6x + 1 = 0 touch each other at the point (1, 2)
The tangent to the curve y = e2x at the point (0, 1) meets x-axis at ______.
The slope of the tangent to the curve x = a sin t, y = a{cot t + log(tan `"t"/2`)} at the point ‘t’ is ____________.
The points at which the tangent passes through the origin for the curve y = 4x3 – 2x5 are
Let `y = f(x)` be the equation of the curve, then equation of normal is
The normals to the curve x = a(θ + sinθ), y = a(1 – cosθ) at the points θ = (2n + 1)π, n∈I are all ______.
If β is one of the angles between the normals to the ellipse, x2 + 3y2 = 9 at the points `(3cosθ, sqrt(3) sinθ)` and `(-3sinθ, sqrt(3) cos θ); θ ∈(0, π/2)`; then `(2 cot β)/(sin 2θ)` is equal to ______.
