Advertisements
Advertisements
Question
Find the equation of the tangent line to the curve y = x2 + 4x − 16 which is parallel to the line 3x − y + 1 = 0 ?
Advertisements
Solution
Let (x0, y0) be the point of intersection of both the curve and the tangent.
\[y = x^2 + 4x - 16\]
\[\text { Since }, \left( x_0 , y_0 \right) \text { lies on curve . Therefore }\]
\[ y_0 = {x_0}^2 + 4 x_0 - 16 . . . \left( 1 \right)\]
\[\text { Now,} y = x^2 + 4x - 16\]
\[ \Rightarrow \frac{dy}{dx} = 2x + 4\]
\[\text { Slope of tangent} = \left( \frac{dy}{dx} \right)_\left( x_0 , y_0 \right) =2 x_0 +4\]
\[\text { Given that The tangent is parallel to the line So,}\]
\[\text { Slope of tangent=slope of the given line}\]
\[2 x_0 + 4 = 3\]
\[ \Rightarrow 2 x_0 = - 1\]
\[ \Rightarrow x_0 = \frac{- 1}{2}\]
\[\text { From} (1),\]
\[ y_0 = \frac{1}{4} - 2 - 16 = \frac{- 71}{4}\]
\[\text { Now, slope of tangent},m=3\]
\[\left( x_0 , y_0 \right) = \left( \frac{- 1}{2}, \frac{- 71}{4} \right)\]
\[\text { Equation of tangent is }\]
\[y - y_0 = m \left( x - x_0 \right)\]
\[ \Rightarrow y + \frac{71}{4} = 3\left( x + \frac{1}{2} \right)\]
\[ \Rightarrow \frac{4y + 71}{4} = 3\left( \frac{2x + 1}{2} \right)\]
\[ \Rightarrow 4y + 71 = 12x + 6\]
\[ \Rightarrow 12x - 4y - 65 = 0\]
APPEARS IN
RELATED QUESTIONS
Find the slope of the tangent to the curve y = (x -1)/(x - 2), x != 2 at x = 10.
Find the slope of the tangent and the normal to the following curve at the indicted point \[y = \sqrt{x^3} \text { at } x = 4\] ?
Find the values of a and b if the slope of the tangent to the curve xy + ax + by = 2 at (1, 1) is 2 ?
Find a point on the curve y = x3 − 3x where the tangent is parallel to the chord joining (1, −2) and (2, 2) ?
Find the point on the curve y = x2 where the slope of the tangent is equal to the x-coordinate of the point ?
At what points on the circle x2 + y2 − 2x − 4y + 1 = 0, the tangent is parallel to x-axis?
At what points on the curve y = x2 − 4x + 5 is the tangent perpendicular to the line 2y + x = 7?
Find the points on the curve \[\frac{x^2}{9} + \frac{y^2}{16} = 1\] at which the tangent is parallel to y-axis ?
Find the equation of the tangent to the curve \[\sqrt{x} + \sqrt{y} = a\] at the point \[\left( \frac{a^2}{4}, \frac{a^2}{4} \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point y2 = 4ax at \[\left( \frac{a}{m^2}, \frac{2a}{m} \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( x_0 , y_0 \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( \sqrt{2}a, b \right)\] ?
Find the equation of the tangent to the curve x = θ + sin θ, y = 1 + cos θ at θ = π/4 ?
Find the equation of the tangent and the normal to the following curve at the indicated points x = θ + sinθ, y = 1 + cosθ at θ = \[\frac{\pi}{2}\] ?
Find the equation of the tangent and the normal to the following curve at the indicated points:
x = 3cosθ − cos3θ, y = 3sinθ − sin3θ?
Find the equation of the normal to the curve ay2 = x3 at the point (am2, am3) ?
The equation of the tangent at (2, 3) on the curve y2 = ax3 + b is y = 4x − 5. Find the values of a and b ?
Find the equations of all lines having slope 2 and that are tangent to the curve \[y = \frac{1}{x - 3}, x \neq 3\] ?
Find the angle of intersection of the following curve x2 + y2 − 4x − 1 = 0 and x2 + y2 − 2y − 9 = 0 ?
Show that the following set of curve intersect orthogonally x2 + 4y2 = 8 and x2 − 2y2 = 4 ?
If the straight line xcos \[\alpha\] +y sin \[\alpha\] = p touches the curve \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\] then prove that a2cos2 \[\alpha\] \[-\] b2sin2 \[\alpha\] = p2 ?
Find the point on the curve y = x2 − 2x + 3, where the tangent is parallel to x-axis ?
Write the angle made by the tangent to the curve x = et cos t, y = et sin t at \[t = \frac{\pi}{4}\] with the x-axis ?
Write the angle between the curves y2 = 4x and x2 = 2y − 3 at the point (1, 2) ?
Write the equation of the normal to the curve y = cos x at (0, 1) ?
The equation of the normal to the curve y = x + sin x cos x at x = `π/2` is ___________ .
The equation of the normal to the curve 3x2 − y2 = 8 which is parallel to x + 3y = 8 is ____________ .
At what point the slope of the tangent to the curve x2 + y2 − 2x − 3 = 0 is zero
The curves y = aex and y = be−x cut orthogonally, if ___________ .
The point on the curve y = 6x − x2 at which the tangent to the curve is inclined at π/4 to the line x + y= 0 is __________ .
The normal at the point (1, 1) on the curve 2y + x2 = 3 is _____________ .
Find the equation of tangents to the curve y = cos(x + y), –2π ≤ x ≤ 2π that are parallel to the line x + 2y = 0.
Find the equation of all the tangents to the curve y = cos(x + y), –2π ≤ x ≤ 2π, that are parallel to the line x + 2y = 0.
The slope of tangent to the curve x = t2 + 3t – 8, y = 2t2 – 2t – 5 at the point (2, –1) is ______.
The two curves x3 – 3xy2 + 2 = 0 and 3x2y – y3 – 2 = 0 intersect at an angle of ______.
The equation of normal to the curve y = tanx at (0, 0) is ______.
The point on the curves y = (x – 3)2 where the tangent is parallel to the chord joining (3, 0) and (4, 1) is ____________.
If the tangent to the curve y = x + siny at a point (a, b) is parallel to the line joining `(0, 3/2)` and `(1/2, 2)`, then ______.
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 m be the slope of a tangent to the curve e2y = 1 + 4x2, then ______.
